diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index 8d2fa800d..1f8ce209a 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -11,7 +11,7 @@ repos:
     -   id: check-symlinks
     -   id: mixed-line-ending
 -   repo: https://github.com/pre-commit/mirrors-clang-format
-    rev: v18.1.8
+    rev: v19.1.0
     hooks:
     -   id: clang-format
 ci:
diff --git a/extern/sedov/sedov.ipynb b/extern/sedov/sedov.ipynb
index c227c84ca..b2995bd0e 100644
--- a/extern/sedov/sedov.ipynb
+++ b/extern/sedov/sedov.ipynb
@@ -1,177 +1,1204 @@
 {
-  "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 15,
-   "metadata": {},
-   "outputs": [
+  "cells" : [
     {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "yt : [INFO     ] 2022-06-07 16:08:58,456 Parameters: current_time              = 1.0\n",
-      "yt : [INFO     ] 2022-06-07 16:08:58,457 Parameters: domain_dimensions         = [128 128 128]\n",
-      "yt : [INFO     ] 2022-06-07 16:08:58,457 Parameters: domain_left_edge          = [0. 0. 0.]\n",
-      "yt : [INFO     ] 2022-06-07 16:08:58,458 Parameters: domain_right_edge         = [1.2 1.2 1.2]\n"
-     ]
-    }
-   ],
-   "source": [
-    "import yt\n",
-    "ds = yt.load(\"../../tests/blast128_pltfile\")\n",
-    "my_sphere = ds.sphere([0., 0., 0.], 1.2)\n",
-    "plot = yt.ProfilePlot(my_sphere, \"radius\", [(\"boxlib\", \"gasDensity\")], weight_field=\"ones\")\n",
-    "plot.set_log((\"boxlib\", \"gasDensity\"), False)\n",
-    "plot.set_log(\"radius\", False);"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 16,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f6954e5f520>"
+      "cell_type" : "code",
+      "execution_count" : 15,
+      "metadata" : {},
+      "outputs" : [ {
+        "name" : "stderr",
+        "output_type" : "stream",
+        "text" : [
+          "yt : [INFO     ] 2022-06-07 16:08:58,456 Parameters: current_time   "
+          "           = 1.0\n",
+          "yt : [INFO     ] 2022-06-07 16:08:58,457 Parameters: "
+          "domain_dimensions         = [128 128 128]\n",
+          "yt : [INFO     ] 2022-06-07 16:08:58,457 Parameters: "
+          "domain_left_edge          = [0. 0. 0.]\n",
+          "yt : [INFO     ] 2022-06-07 16:08:58,458 Parameters: "
+          "domain_right_edge         = [1.2 1.2 1.2]\n"
+        ]
+      } ],
+      "source" : [
+        "import yt\n", "ds = yt.load(\"../../tests/blast128_pltfile\")\n",
+        "my_sphere = ds.sphere([0., 0., 0.], 1.2)\n",
+        "plot = yt.ProfilePlot(my_sphere, \"radius\", [(\"boxlib\", "
+        "\"gasDensity\")], weight_field=\"ones\")\n",
+        "plot.set_log((\"boxlib\", \"gasDensity\"), False)\n",
+        "plot.set_log(\"radius\", False);"
       ]
-     },
-     "execution_count": 16,
-     "metadata": {},
-     "output_type": "execute_result"
     },
     {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 432x288 with 1 Axes>"
+      "cell_type" : "code",
+      "execution_count" : 16,
+      "metadata" : {},
+      "outputs" : [
+        {
+          "data" :
+              {"text/plain" : ["<matplotlib.legend.Legend at 0x7f6954e5f520>"]},
+          "execution_count" : 16,
+          "metadata" : {},
+          "output_type" : "execute_result"
+        },
+        {
+          "data" : {
+            "image/png" :
+                "iVBORw0KGgoAAAANSUhEUgAAAX4AAAEHCAYAAACp9y31AAAAOXRFWHRTb2Z0d2"
+                "FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGli"
+                "Lm9yZy/"
+                "YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAAqlElEQVR4nO3de3xU9Z3/"
+                "8ddnJhMS7iA3ATV4Q+4Xo6VeKXjBS5FarVpsbbettn1sd3/b/"
+                "WWLrdvL1tbbbn/"
+                "Wx7bdUtu1VtuyVUS23qoVi4KiXAW5lIoiBJQESYCQhGTy+"
+                "f0xkxBgMjMnmWSSzPv5ePBg5pwz53wPCe988j3f8z3m7oiISO4IZbsBIiLSsRT"
+                "8IiI5RsEvIpJjFPwiIjlGwS8ikmMU/CIiOSYvGwc1s/7Ag8B4wIG/c/"
+                "dXE207aNAgLyoq6rjGiYh0A6tWrSp398GJ1mUl+IEfA8+6+3Vmlg/"
+                "0bGnDoqIiVq5c2XEtExHpBsxse0vrOjz4zawfcBHwOQB3Pwwc7uh2iIjkqmz08"
+                "Y8CyoD/"
+                "NrM1ZvagmfVqvoGZ3WpmK81sZVlZWRaaKCLSfWUj+"
+                "POAqcDP3H0KUAXMa76Bu89392J3Lx48OGEXlYiItFI2+vh3AjvdfUX8/WMcE/"
+                "yp1NXVsXPnTmpqajLeOIGCggJGjhxJJBLJdlNEpB10ePC7+/"
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+                "OxdwAAAABJRU5ErkJggg==",
+            "text/plain" : ["<Figure size 432x288 with 1 Axes>"]
+          },
+          "metadata" : {"needs_background" : "light"},
+          "output_type" : "display_data"
+        }
+      ],
+      "source" : [
+        "# see https://cococubed.com/research_pages/sedov.shtml\n",
+        "# for exact solutions\n", "\n", "import numpy as np\n",
+        "i, x, den, ener, pres, vel, cs = "
+        "np.loadtxt(\"spherical_standard_omega0p00.dat\", unpack=True, "
+        "skiprows=2)\n",
+        "\n", "profile = plot.profiles[0]\n", "r = profile.x\n",
+        "rho = profile[(\"boxlib\",\"gasDensity\")]\n", "\n",
+        "import matplotlib.pyplot as plt\n",
+        "plt.scatter(r, rho, label='simulation')\n",
+        "plt.plot(x, den, label='exact')\n", "\n",
+        "plt.xlabel(r\"radius (cm)\")\n",
+        "plt.ylabel(r\"density (g cm$^{-3}$)\")\n", "plt.legend(loc='best')"
       ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# see https://cococubed.com/research_pages/sedov.shtml\n",
-    "# for exact solutions\n",
-    "\n",
-    "import numpy as np\n",
-    "i, x, den, ener, pres, vel, cs = np.loadtxt(\"spherical_standard_omega0p00.dat\", unpack=True, skiprows=2)\n",
-    "\n",
-    "profile = plot.profiles[0]\n",
-    "r = profile.x\n",
-    "rho = profile[(\"boxlib\",\"gasDensity\")]\n",
-    "\n",
-    "import matplotlib.pyplot as plt\n",
-    "plt.scatter(r, rho, label='simulation')\n",
-    "plt.plot(x, den, label='exact')\n",
-    "\n",
-    "plt.xlabel(r\"radius (cm)\")\n",
-    "plt.ylabel(r\"density (g cm$^{-3}$)\")\n",
-    "plt.legend(loc='best')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "0.0 dimensionless 26844572019.077103 dimensionless\n"
-     ]
     },
     {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 432x288 with 1 Axes>"
+      "cell_type" : "code",
+      "execution_count" : 17,
+      "metadata" : {},
+      "outputs" : [
+        {
+          "name" : "stdout",
+          "output_type" : "stream",
+          "text" : ["0.0 dimensionless 26844572019.077103 dimensionless\n"]
+        },
+        {
+          "data" : {
+            "image/png" :
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+                "Zn99PfCFsfeYBKl6j0lEzgZewRYAwd7ODPUckzGmxBiTaozJMcbkYGvHHGKMKQ"
+                "hMuW5x53dvFrazAEQkFVvz0FY/"
+                "1thQ7hzTTuBSABHJxRYCRX6t0rtmA7fZrxLqA5QYY/"
+                "YGuihPiEgbbOOzjTTGbHR7xUD3eHvQU34Ftm9YW4BH7PP+jO1DBGy/"
+                "pO8Dm4HvgHaBrtkLx/QZtju2V9h/Zge6Zk+PyWnZ+QT51UFu/"
+                "jsJtmautcAqYHiga/bCMeUBX2G7cmgFcHmga67neKYBe4FKbGdmo4G7gLsc/"
+                "o0m2I93VRP5vavvmF7FNlT/"
+                "qc+"
+                "HAne2q3cMK6VUCGuqzUFKKaW8QENAKaVCmIaAUkqFMA0BpZQKYRoCSikVwjQEl"
+                "FIqhGkIKKVUCPt/w8MPACkgX4YAAAAASUVORK5CYII=",
+            "text/plain" : ["<Figure size 432x288 with 1 Axes>"]
+          },
+          "metadata" : {"needs_background" : "light"},
+          "output_type" : "display_data"
+        }
+      ],
+      "source" : [
+        "# interpolate exact solution\n",
+        "exact_at_sim = np.interp(r, x, den)\n",
+        "diff = np.abs(rho - exact_at_sim)\n",
+        "rel_diff = diff / exact_at_sim\n", "\n",
+        "print(np.min(rel_diff), np.max(rel_diff))\n",
+        "plt.plot(r, rel_diff, label='density')\n", "plt.yscale('log')\n",
+        "plt.ylim(1e-5, 1e10)\n", "plt.title('relative error')\n",
+        "plt.legend(loc='upper right')\n", "plt.show()"
       ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# interpolate exact solution\n",
-    "exact_at_sim = np.interp(r, x, den)\n",
-    "diff = np.abs(rho - exact_at_sim)\n",
-    "rel_diff = diff / exact_at_sim\n",
-    "\n",
-    "print(np.min(rel_diff), np.max(rel_diff))\n",
-    "plt.plot(r, rel_diff, label='density')\n",
-    "plt.yscale('log')\n",
-    "plt.ylim(1e-5, 1e10)\n",
-    "plt.title('relative error')\n",
-    "plt.legend(loc='upper right')\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 18,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "0.0 dimensionless 1.6895137149977217 dimensionless\n"
-     ]
     },
     {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 432x288 with 1 Axes>"
+      "cell_type" : "code",
+      "execution_count" : 18,
+      "metadata" : {},
+      "outputs" : [
+        {
+          "name" : "stdout",
+          "output_type" : "stream",
+          "text" : ["0.0 dimensionless 1.6895137149977217 dimensionless\n"]
+        },
+        {
+          "data" : {
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+                "aD1HT1Z7mVeM8asNsYsTUnpuiplT+h9AqHjnq+NpH9KAv/"
+                "22k6vZ2Pkl1ZrV5DyiVy7jcMn6zjd4Ah2KF7zJglsAUaKyDARiQNuBlb5IigRm"
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+                "KukaOVtVrElA+"
+                "kZPp6lI8UBH6M4Q8nSL6AvARkCsiJSJyhzHGAdwNrAXygZXGmF2+"
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+                "nD+GRoeTh9/u2SBxy8wgrRmk1LksWTbC/2MCKtQMS0/"
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+                "FpR1e9v8UtedwlouQqlzWTIJ+"
+                "H9MQC8EoWjykD6kJ8fz9s5j3drO0eyk8HiNjgco1Q5LJgGl2hMdJcwem8m6gjL"
+                "ONDZ7vN2BitM0Opw6M0j5nCdP8rM6SyYBf3cHaTsgdM0d258zTc18sMfzLqF89"
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+                "urfBofS0XoVTHLJkE/"
+                "D4moE2BkDZ9RBopvWJZs7O0y3VPnm7keHWDPkNAqQ5YMgnomIDqTGx0FFeNzuT"
+                "d3cdpdHR+236BPkNAqU5ZMgko1ZW54+zU1DvYtK/"
+                "zLqHdmgSUX4X+HFFNAiokXZKdji0+hre+7LxLKL+"
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+                "8tP09is5SKU6kzEJQEVPhJio/laXj/W7jxGs/"
+                "P8Abp8HRRWqksRlwT0HoHwMmdsf06cbuTTA+"
+                "d3CeWXVhMXHcXwjKQgRKYigRaQUyrIZuZmkBAbxVvtdAntLq0mu18ysdH6a658"
+                "K5w+S0bcX0cY/"
+                "ewUrgfRz8zpx9s7j+Fs0yVUcKxGu4KU6oIlk4A+WUx1x5xxdspqGth2+"
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+                "tHXHTUOeWldVBYKc9YMgko1R22hFhm5KTz1s7Ss11CmgSU8kzEJQHRUYGwNGds"
+                "f0qr6tlRUglAQWkNmb3j6ZsUF9zAVFgLg8lBkZcEVHiaNSqT2Gg5+"
+                "xD63foMAeVH4fRhMvKSQPj87FQrKYmxXJKdzpqdpTQ4mtlXXqtJQCkPRF4SUGF"
+                "rzlg7xSfP8Pr2ozQ1G/"
+                "LsOjNIqa4ENAmISJKIbBWRrwfyuOfEEKwDK7+"
+                "7arSd6Cjhsff3AmjhOKU84FESEJGnRKRMRHa2WT5bRApFpEhE7vdgVz8BVvYkU"
+                "KW60jcpjouHp1Fy6gxxMVEMS9dyEUp1xdOWwNPA7NYLRCQaeAKYA4wGFovIaBE"
+                "ZJyJvtPnqJyJXAbuBMh/G3216n0B4mzPODkBupo0YLReh/"
+                "CwcagfFeLKSMWaDiAxts3gaUGSM2Q8gIiuA64wxDwHndfeIyEwgCVfCOCMia4w"
+                "xnT8gVqluunq0nX9/"
+                "bafeKaz8Kpw+"
+                "THqUBDowEChu9boEuLCjlY0xPwcQkSVARUcJQESWAksBBg8e7EV47QunqV3qfB"
+                "m2eP70rck6KKyUh7xJAj1ijHm6i/"
+                "eXiUgpMC8uLm5yYKJS4eSaMfZgh6BUyPCm0/"
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+                "CMgVkRIRucMY4wDuBtYC+cBKY8wuXwTl15aAz/"
+                "eolIpUJgymB3k6O2hxB8vXAGt8GhGulgAwLzs729e7Vkopr4XTh0lLTqT2Z0tA"
+                "KaXUVyyZBJRSSgWGJZOADgwrpVRgWDIJaHeQUkoFhiWTgLYElFIqMCyZBLxtCb"
+                "yz6xjLPzzg46iUUupcoT9BNAhlIwLhnd3H+"
+                "WjfiWCHoZQKV2E0R9SSLQGllFKBYckkoGMCSikVGJZMAjo7SCmlAsOSSUAppVR"
+                "gaBJQSqkeCoP6cZoElFKqu8LpCYWWTAK+"
+                "GBgOhxKvSinlb5ZMAt4ODIdPjlZKKf+"
+                "yZBJQSikVGBGXBEQfMqyUUmdFXBJQSilfMWFQPUiTgFJKdVM4dShYMglo2Qill"
+                "AoMSyYBLRuhlFKBYckk4Auh31OnlFL+F5ZJIJz665RSyp/"
+                "CMgkopVRAhEGXgyYBpZTqpnDqbNAkoJRSEUyTgFJKRbCAJQERmSkiG0XkSRGZG"
+                "ajjKqWU6phHSUBEnhKRMhHZ2Wb5bBEpFJEiEbm/"
+                "i90YoBZIAEp6Fq5SSilfivFwvaeBx4FnWxaISDTwBHAVrov6FhFZBUQDD7XZ/"
+                "nZgozHmAxHJBB4BbvEu9M519DgBfc6AUspXwuFqIp5eFEVkKPCGMWas+/"
+                "XFwAPGmGvcr38KYIxpmwDa7icOeN4Yc2MH7y8Flrpf5gKFHgX4lXSgopvbWJ2e"
+                "U2jQcwoNkXBOQ4wxGZ5s6GlLoD0DgeJWr0uACztaWUS+"
+                "AVwDpOJqVbTLGLMMWNbToERkqzFmSk+"
+                "3tyI9p9Cg5xQa9JzO5U0S6BZjzKvAq4E6nlJKqa55MzvoCDCo1ess9zKllFIhw"
+                "psksAUYKSLD3P38NwOrfBOWV3rclWRhek6hQc8pNOg5teLRwLCIvADMxDX4cBz"
+                "4hTFmuYjMBX6Ha0bQU8aYB3saiFJKqcDzeHaQUkqp8KNlI5RSKoKFbBLo6m5lE"
+                "YkXkRfd73/ivs/B0jw4p/tEZLeIfCEi74vIkGDE2R2e3lUuIjeIiBERy0/"
+                "d8+ScRGSh+2e1S0SeD3SM3eXB795gEfmniHzu/"
+                "v2bG4w4u6OjSget3hcRecx9zl+IyKRAx9hdHpzTLe5z+VJENovI+"
+                "C53aowJuS9cYxD7gOFAHLADGN1mnbuAJ93f3wy8GOy4fXBOVwCJ7u+/"
+                "Hw7n5F7PBmwAPgamBDtuH/ycRgKfA33cr/"
+                "sFO24fnNMy4Pvu70cDB4MdtwfnNQOYBOzs4P25wFu4KkNfBHwS7Jh9cE7TW/"
+                "3ezfHknEK1JTANKDLG7DfGNAIrgOvarHMd8Iz7+5eBK0Us/cyxLs/"
+                "JGPNPY0yd++XHuKblWpknPyeAXwG/"
+                "BuoDGVwPeXJOdwJPGGNOARhjygIcY3d5ck4G6O3+"
+                "PgU4GsD4esQYswE42ckq1wHPGpePgVQR6R+Y6Hqmq3Myxmxu+"
+                "b3Dw2tEqCaB9u5WHtjROsYYB1AFpAUkup7x5JxauwPXpxgr6/"
+                "Kc3E3wQcaYNwMZmBc8+TnlADkisklEPhaR2QGLrmc8OacHgG+"
+                "JSAmwBvhBYELzq+7+zYUaj64RAbtjWPmOiHwLmAJcHuxYvCEiUbiKCS4Jcii+"
+                "FoOrS2gmrk9iG0RknDGmMphBeWkx8LQx5n/"
+                "cdcP+JiJjjTHOYAemziciV+"
+                "BKApd2tW6otgQ8uVv57DoiEoOrCXsiINH1jEd3YIvILODnwHxjTEOAYuuprs7J"
+                "BowF1ovIQVz9sqssPjjsyc+"
+                "pBFhljGkyxhwA9uBKClblyTndAawEMMZ8hKskfHpAovOfsKx6ICIXAH8BrjPGd"
+                "HnNC9Uk4MndyquA77i/"
+                "vxFYZ9yjJRbV5TmJyETgz7gSgNX7maGLczLGVBlj0o0xQ40xQ3H1Yc43xmwNTr"
+                "ge8eR37zVcrQBEJB1X99D+"
+                "AMbYXZ6c02HgSgARGYUrCZQHNErfWwXc6p4ldBFQZYwpDXZQ3hCRwbhqtH3bGL"
+                "PHo42CPdrtxSj5XFyfsPYBP3cv+w9cFxFw/"
+                "ZK+BBQBnwLDgx2zD87pPVx3bG93f60KdszenlObdddj8dlBHv6cBFc3127gS+"
+                "DmYMfsg3MaDWzCNXNoO3B1sGP24JxeAEqBJlytszuA7wHfa/"
+                "VzesJ9zl+GyO9eV+f0F+BUq2vE1q72qXcMK6VUBAvV7iCllFI+"
+                "oElAKaUimCYBpZSKYJoElFIqgmkSUEqpCKZJQCmlIpgmAaWUimD/"
+                "D3Bw1ezxA5FWAAAAAElFTkSuQmCC",
+            "text/plain" : ["<Figure size 432x288 with 1 Axes>"]
+          },
+          "metadata" : {"needs_background" : "light"},
+          "output_type" : "display_data"
+        }
+      ],
+      "source" : [
+        "print(np.min(diff), np.max(diff))\n",
+        "plt.plot(r, diff, label='density')\n", "plt.yscale('log')\n",
+        "plt.ylim(1e-5, 10)\n", "plt.title('absolute error')\n",
+        "plt.legend(loc='upper left')\n", "plt.show()"
       ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
     }
-   ],
-   "source": [
-    "print(np.min(diff), np.max(diff))\n",
-    "plt.plot(r, diff, label='density')\n",
-    "plt.yscale('log')\n",
-    "plt.ylim(1e-5, 10)\n",
-    "plt.title('absolute error')\n",
-    "plt.legend(loc='upper left')\n",
-    "plt.show()"
-   ]
-  }
- ],
- "metadata": {
-  "interpreter": {
-   "hash": "5d21aa510141f516aa4589cf93de6e254d300587727b9d0aa72219f1336bba5f"
-  },
-  "kernelspec": {
-   "display_name": "Python 3.9.7 ('base')",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.9.7"
+  ],
+  "metadata" : {
+    "interpreter" : {
+      "hash" :
+          "5d21aa510141f516aa4589cf93de6e254d300587727b9d0aa72219f1336bba5f"
+    },
+    "kernelspec" : {
+      "display_name" : "Python 3.9.7 ('base')",
+      "language" : "python",
+      "name" : "python3"
+    },
+    "language_info" : {
+      "codemirror_mode" : {"name" : "ipython", "version" : 3},
+      "file_extension" : ".py",
+      "mimetype" : "text/x-python",
+      "name" : "python",
+      "nbconvert_exporter" : "python",
+      "pygments_lexer" : "ipython3",
+      "version" : "3.9.7"
+    },
+    "orig_nbformat" : 4
   },
-  "orig_nbformat": 4
- },
- "nbformat": 4,
- "nbformat_minor": 2
+  "nbformat" : 4,
+  "nbformat_minor" : 2
 }
diff --git a/src/QuokkaSimulation.hpp b/src/QuokkaSimulation.hpp
index 7177e675d..a16bca36a 100644
--- a/src/QuokkaSimulation.hpp
+++ b/src/QuokkaSimulation.hpp
@@ -221,8 +221,8 @@ template <typename problem_t> class QuokkaSimulation : public AMRSimulation<prob
 	void addFluxArrays(std::array<amrex::MultiFab, AMREX_SPACEDIM> &dstfluxes, std::array<amrex::MultiFab, AMREX_SPACEDIM> &srcfluxes, int srccomp,
 			   int dstcomp);
 
-	auto expandFluxArrays(std::array<amrex::FArrayBox, AMREX_SPACEDIM> &fluxes, int nstartNew,
-			      int ncompNew) -> std::array<amrex::FArrayBox, AMREX_SPACEDIM>;
+	auto expandFluxArrays(std::array<amrex::FArrayBox, AMREX_SPACEDIM> &fluxes, int nstartNew, int ncompNew)
+	    -> std::array<amrex::FArrayBox, AMREX_SPACEDIM>;
 
 	void printCoordinates(int lev, const amrex::IntVect &cell_idx);
 
@@ -258,11 +258,11 @@ template <typename problem_t> class QuokkaSimulation : public AMRSimulation<prob
 				    amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> dx)
 	    -> std::tuple<std::array<amrex::FArrayBox, AMREX_SPACEDIM>, std::array<amrex::FArrayBox, AMREX_SPACEDIM>>;
 
-	auto computeHydroFluxes(amrex::MultiFab const &consVar, int nvars,
-				int lev) -> std::pair<std::array<amrex::MultiFab, AMREX_SPACEDIM>, std::array<amrex::MultiFab, AMREX_SPACEDIM>>;
+	auto computeHydroFluxes(amrex::MultiFab const &consVar, int nvars, int lev)
+	    -> std::pair<std::array<amrex::MultiFab, AMREX_SPACEDIM>, std::array<amrex::MultiFab, AMREX_SPACEDIM>>;
 
-	auto computeFOHydroFluxes(amrex::MultiFab const &consVar, int nvars,
-				  int lev) -> std::pair<std::array<amrex::MultiFab, AMREX_SPACEDIM>, std::array<amrex::MultiFab, AMREX_SPACEDIM>>;
+	auto computeFOHydroFluxes(amrex::MultiFab const &consVar, int nvars, int lev)
+	    -> std::pair<std::array<amrex::MultiFab, AMREX_SPACEDIM>, std::array<amrex::MultiFab, AMREX_SPACEDIM>>;
 
 	template <FluxDir DIR>
 	void fluxFunction(amrex::Array4<const amrex::Real> const &consState, amrex::FArrayBox &x1Flux, amrex::FArrayBox &x1FluxDiffusive,
@@ -1381,8 +1381,8 @@ void QuokkaSimulation<problem_t>::addFluxArrays(std::array<amrex::MultiFab, AMRE
 }
 
 template <typename problem_t>
-auto QuokkaSimulation<problem_t>::expandFluxArrays(std::array<amrex::FArrayBox, AMREX_SPACEDIM> &fluxes, const int nstartNew,
-						   const int ncompNew) -> std::array<amrex::FArrayBox, AMREX_SPACEDIM>
+auto QuokkaSimulation<problem_t>::expandFluxArrays(std::array<amrex::FArrayBox, AMREX_SPACEDIM> &fluxes, const int nstartNew, const int ncompNew)
+    -> std::array<amrex::FArrayBox, AMREX_SPACEDIM>
 {
 	BL_PROFILE("QuokkaSimulation::expandFluxArrays()");
 
diff --git a/src/hydro/EOS.hpp b/src/hydro/EOS.hpp
index e52648712..a8c5a5e60 100644
--- a/src/hydro/EOS.hpp
+++ b/src/hydro/EOS.hpp
@@ -48,12 +48,12 @@ template <typename problem_t> class EOS
 	ComputeEintFromTgas(amrex::Real rho, amrex::Real Tgas, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {}) -> amrex::Real;
 
 	[[nodiscard]] AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE static auto
-	ComputeEintFromPres(amrex::Real rho, amrex::Real Pressure,
-			    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {}) -> amrex::Real;
+	ComputeEintFromPres(amrex::Real rho, amrex::Real Pressure, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {})
+	    -> amrex::Real;
 
 	[[nodiscard]] AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE static auto
-	ComputeEintTempDerivative(amrex::Real rho, amrex::Real Tgas,
-				  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {}) -> amrex::Real;
+	ComputeEintTempDerivative(amrex::Real rho, amrex::Real Tgas, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {})
+	    -> amrex::Real;
 
 	[[nodiscard]] AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE static auto
 	ComputeOtherDerivatives(amrex::Real rho, amrex::Real P, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {});
@@ -62,8 +62,8 @@ template <typename problem_t> class EOS
 	ComputePressure(amrex::Real rho, amrex::Real Eint, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {}) -> amrex::Real;
 
 	[[nodiscard]] AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE static auto
-	ComputeSoundSpeed(amrex::Real rho, amrex::Real Pressure,
-			  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {}) -> amrex::Real;
+	ComputeSoundSpeed(amrex::Real rho, amrex::Real Pressure, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars = {})
+	    -> amrex::Real;
 
       private:
 	static constexpr amrex::Real gamma_ = EOS_Traits<problem_t>::gamma;
@@ -72,9 +72,9 @@ template <typename problem_t> class EOS
 };
 
 template <typename problem_t>
-AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto
-EOS<problem_t>::ComputeTgasFromEint(amrex::Real rho, amrex::Real Eint,
-				    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars) -> amrex::Real
+AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto EOS<problem_t>::ComputeTgasFromEint(amrex::Real rho, amrex::Real Eint,
+										  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars)
+    -> amrex::Real
 {
 	// return temperature for an ideal gas given density and internal energy
 	amrex::Real Tgas = NAN;
@@ -114,9 +114,9 @@ EOS<problem_t>::ComputeTgasFromEint(amrex::Real rho, amrex::Real Eint,
 }
 
 template <typename problem_t>
-AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto
-EOS<problem_t>::ComputeEintFromTgas(amrex::Real rho, amrex::Real Tgas,
-				    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars) -> amrex::Real
+AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto EOS<problem_t>::ComputeEintFromTgas(amrex::Real rho, amrex::Real Tgas,
+										  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars)
+    -> amrex::Real
 {
 	// return internal energy density given density and temperature
 	amrex::Real Eint = NAN;
@@ -157,9 +157,9 @@ EOS<problem_t>::ComputeEintFromTgas(amrex::Real rho, amrex::Real Tgas,
 }
 
 template <typename problem_t>
-AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto
-EOS<problem_t>::ComputeEintFromPres(amrex::Real rho, amrex::Real Pressure,
-				    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars) -> amrex::Real
+AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto EOS<problem_t>::ComputeEintFromPres(amrex::Real rho, amrex::Real Pressure,
+										  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars)
+    -> amrex::Real
 {
 	// return internal energy density given density and pressure
 	amrex::Real Eint = NAN;
@@ -297,8 +297,9 @@ EOS<problem_t>::ComputeOtherDerivatives(const amrex::Real rho, const amrex::Real
 }
 
 template <typename problem_t>
-AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto
-EOS<problem_t>::ComputePressure(amrex::Real rho, amrex::Real Eint, std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars) -> amrex::Real
+AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto EOS<problem_t>::ComputePressure(amrex::Real rho, amrex::Real Eint,
+									      std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars)
+    -> amrex::Real
 {
 	// return pressure for an ideal gas
 	amrex::Real P = NAN;
@@ -341,9 +342,9 @@ EOS<problem_t>::ComputePressure(amrex::Real rho, amrex::Real Eint, std::optional
 }
 
 template <typename problem_t>
-AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto
-EOS<problem_t>::ComputeSoundSpeed(amrex::Real rho, amrex::Real Pressure,
-				  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars) -> amrex::Real
+AMREX_FORCE_INLINE AMREX_GPU_HOST_DEVICE auto EOS<problem_t>::ComputeSoundSpeed(amrex::Real rho, amrex::Real Pressure,
+										std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const &massScalars)
+    -> amrex::Real
 {
 	// return sound speed for an ideal gas
 	amrex::Real cs = NAN;
diff --git a/src/hydro/LLF.hpp b/src/hydro/LLF.hpp
index f0c5b54c4..f6fa06564 100644
--- a/src/hydro/LLF.hpp
+++ b/src/hydro/LLF.hpp
@@ -13,8 +13,8 @@ namespace quokka::Riemann
 {
 // Local Lax-Friedrichs (LLF) / Rusanov solver
 template <typename problem_t, int N_scalars, int N_mscalars, int fluxdim>
-AMREX_FORCE_INLINE AMREX_GPU_DEVICE auto LLF(quokka::HydroState<N_scalars, N_mscalars> const &sL,
-					     quokka::HydroState<N_scalars, N_mscalars> const &sR) -> quokka::valarray<double, fluxdim>
+AMREX_FORCE_INLINE AMREX_GPU_DEVICE auto LLF(quokka::HydroState<N_scalars, N_mscalars> const &sL, quokka::HydroState<N_scalars, N_mscalars> const &sR)
+    -> quokka::valarray<double, fluxdim>
 {
 	// Toro (Eq. 10.56)
 	const amrex::Real Sp = std::max(std::abs(sL.u) + sL.cs, std::abs(sR.u) + sR.cs);
diff --git a/src/hydro/NSCBC_inflow.hpp b/src/hydro/NSCBC_inflow.hpp
index 02afaa5ad..19b308fae 100644
--- a/src/hydro/NSCBC_inflow.hpp
+++ b/src/hydro/NSCBC_inflow.hpp
@@ -21,10 +21,11 @@ namespace NSCBC
 namespace detail
 {
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto
-dQ_dx_inflow_x1_lower(quokka::valarray<Real, HydroSystem<problem_t>::nvar_> const &Q, quokka::valarray<Real, HydroSystem<problem_t>::nvar_> const &dQ_dx_data,
-		      const Real T_t, const Real u_t, const Real v_t, const Real w_t, amrex::GpuArray<Real, HydroSystem<problem_t>::nscalars_> const &s_t,
-		      const Real L_x) -> quokka::valarray<Real, HydroSystem<problem_t>::nvar_>
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto dQ_dx_inflow_x1_lower(quokka::valarray<Real, HydroSystem<problem_t>::nvar_> const &Q,
+							       quokka::valarray<Real, HydroSystem<problem_t>::nvar_> const &dQ_dx_data, const Real T_t,
+							       const Real u_t, const Real v_t, const Real w_t,
+							       amrex::GpuArray<Real, HydroSystem<problem_t>::nscalars_> const &s_t, const Real L_x)
+    -> quokka::valarray<Real, HydroSystem<problem_t>::nvar_>
 {
 	// return dQ/dx corresponding to subsonic inflow on the x1 lower boundary
 	// (This is only necessary for continuous inflows, i.e., where a shock or contact discontinuity is not desired.)
diff --git a/src/hydro/NSCBC_outflow.hpp b/src/hydro/NSCBC_outflow.hpp
index 16d9966df..c081b11cf 100644
--- a/src/hydro/NSCBC_outflow.hpp
+++ b/src/hydro/NSCBC_outflow.hpp
@@ -206,8 +206,8 @@ AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto transverse_zdir_dQ_data(const amrex::In
 }
 
 template <typename problem_t, FluxDir DIR>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto
-permute_vel(quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_> const &Q) -> quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_>
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto permute_vel(quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_> const &Q)
+    -> quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_>
 {
 	// with normal direction DIR, permutes the velocity components so that
 	//  u, v, w are the normal and transverse components, respectively.
@@ -232,8 +232,8 @@ permute_vel(quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_> const &
 }
 
 template <typename problem_t, FluxDir DIR>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto
-unpermute_vel(quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_> const &Q) -> quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_>
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto unpermute_vel(quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_> const &Q)
+    -> quokka::valarray<amrex::Real, HydroSystem<problem_t>::nvar_>
 {
 	// with normal direction DIR, un-permutes the velocity components so that
 	//  u, v, w are the normal and transverse components *prior to calling permute_vel*.
diff --git a/src/hydro/hydro_system.hpp b/src/hydro/hydro_system.hpp
index 01a8b5a76..02b374e0f 100644
--- a/src/hydro/hydro_system.hpp
+++ b/src/hydro/hydro_system.hpp
@@ -291,8 +291,8 @@ template <typename problem_t> auto HydroSystem<problem_t>::CheckStatesValid(amre
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputePrimVars(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-										 int k) -> quokka::valarray<amrex::Real, nvar_>
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputePrimVars(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> quokka::valarray<amrex::Real, nvar_>
 {
 	// convert to primitive vars
 	const auto rho = cons(i, j, k, density_index);
@@ -324,8 +324,8 @@ AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputePrimVars
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto
-HydroSystem<problem_t>::ComputeConsVars(quokka::valarray<amrex::Real, nvar_> const &prim) -> quokka::valarray<amrex::Real, nvar_>
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeConsVars(quokka::valarray<amrex::Real, nvar_> const &prim)
+    -> quokka::valarray<amrex::Real, nvar_>
 {
 	// convert to conserved vars
 	Real const rho = prim[0];
@@ -347,8 +347,8 @@ HydroSystem<problem_t>::ComputeConsVars(quokka::valarray<amrex::Real, nvar_> con
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputePressure(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-										 int k) -> amrex::Real
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputePressure(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> amrex::Real
 {
 	const auto rho = cons(i, j, k, density_index);
 	const auto px = cons(i, j, k, x1Momentum_index);
@@ -372,8 +372,8 @@ AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputePressure
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeSoundSpeed(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-										   int k) -> amrex::Real
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeSoundSpeed(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> amrex::Real
 {
 	const auto rho = cons(i, j, k, density_index);
 	const auto px = cons(i, j, k, x1Momentum_index);
@@ -394,8 +394,8 @@ AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeSoundSpe
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocityX1(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-										   int k) -> amrex::Real
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocityX1(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> amrex::Real
 {
 	amrex::Real const rho = cons(i, j, k, density_index);
 	amrex::Real const vel_x = cons(i, j, k, x1Momentum_index) / rho;
@@ -403,8 +403,8 @@ AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocity
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocityX2(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-										   int k) -> amrex::Real
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocityX2(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> amrex::Real
 {
 	amrex::Real const rho = cons(i, j, k, density_index);
 	amrex::Real const vel_y = cons(i, j, k, x2Momentum_index) / rho;
@@ -412,8 +412,8 @@ AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocity
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocityX3(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-										   int k) -> amrex::Real
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto HydroSystem<problem_t>::ComputeVelocityX3(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> amrex::Real
 {
 	amrex::Real const rho = cons(i, j, k, density_index);
 	amrex::Real const vel_z = cons(i, j, k, x3Momentum_index) / rho;
diff --git a/src/hyperbolic_system.hpp b/src/hyperbolic_system.hpp
index 03f556223..6e9fca156 100644
--- a/src/hyperbolic_system.hpp
+++ b/src/hyperbolic_system.hpp
@@ -65,8 +65,8 @@ template <typename problem_t> class HyperbolicSystem
 		return 0.5 * (sgn(a) + sgn(b)) * std::min(std::abs(a), std::abs(b));
 	}
 
-	[[nodiscard]] AMREX_GPU_DEVICE AMREX_FORCE_INLINE static auto GetMinmaxSurroundingCell(arrayconst_t &q, int i, int j, int k,
-											       int n) -> std::pair<double, double>;
+	[[nodiscard]] AMREX_GPU_DEVICE AMREX_FORCE_INLINE static auto GetMinmaxSurroundingCell(arrayconst_t &q, int i, int j, int k, int n)
+	    -> std::pair<double, double>;
 
 	template <FluxDir DIR>
 	static void ReconstructStatesConstant(amrex::MultiFab const &q, amrex::MultiFab &leftState, amrex::MultiFab &rightState, int nghost, int nvars);
diff --git a/src/linear_advection/AdvectionSimulation.hpp b/src/linear_advection/AdvectionSimulation.hpp
index c26d5afdc..13f4325a5 100644
--- a/src/linear_advection/AdvectionSimulation.hpp
+++ b/src/linear_advection/AdvectionSimulation.hpp
@@ -380,8 +380,8 @@ template <typename problem_t> void AdvectionSimulation<problem_t>::advanceSingle
 }
 
 template <typename problem_t>
-auto AdvectionSimulation<problem_t>::computeFluxes(amrex::MultiFab const &consVar, const int nvars,
-						   const int lev) -> std::array<amrex::MultiFab, AMREX_SPACEDIM>
+auto AdvectionSimulation<problem_t>::computeFluxes(amrex::MultiFab const &consVar, const int nvars, const int lev)
+    -> std::array<amrex::MultiFab, AMREX_SPACEDIM>
 {
 	auto ba = grids[lev];
 	auto dm = dmap[lev];
diff --git a/src/linear_advection/linear_advection.hpp b/src/linear_advection/linear_advection.hpp
index 153e862b5..a7046e539 100644
--- a/src/linear_advection/linear_advection.hpp
+++ b/src/linear_advection/linear_advection.hpp
@@ -72,8 +72,8 @@ void LinearAdvectionSystem<problem_t>::ConservedToPrimitive(amrex::MultiFab cons
 }
 
 template <typename problem_t>
-AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto LinearAdvectionSystem<problem_t>::isStateValid(amrex::Array4<const amrex::Real> const &cons, int i, int j,
-											int k) -> bool
+AMREX_GPU_DEVICE AMREX_FORCE_INLINE auto LinearAdvectionSystem<problem_t>::isStateValid(amrex::Array4<const amrex::Real> const &cons, int i, int j, int k)
+    -> bool
 {
 	// check if cons(i, j, k) is a valid state
 	const auto rho = cons(i, j, k, density_index);
diff --git a/src/math/quadrature.hpp b/src/math/quadrature.hpp
index adfd8c1da..71a3bddf4 100644
--- a/src/math/quadrature.hpp
+++ b/src/math/quadrature.hpp
@@ -22,8 +22,8 @@ AMREX_FORCE_INLINE AMREX_GPU_DEVICE auto kernel_wendland_c2(const amrex::Real r)
 }
 
 template <typename F>
-AMREX_FORCE_INLINE AMREX_GPU_DEVICE auto quad_3d(F &&f, amrex::Real x0, amrex::Real x1, amrex::Real y0, amrex::Real y1, amrex::Real z0,
-						 amrex::Real z1) -> amrex::Real
+AMREX_FORCE_INLINE AMREX_GPU_DEVICE auto quad_3d(F &&f, amrex::Real x0, amrex::Real x1, amrex::Real y0, amrex::Real y1, amrex::Real z0, amrex::Real z1)
+    -> amrex::Real
 {
 	// integrate F over the rectangular domain [x0, y0, z0] -> [x1, y1, z1].
 	auto integrand = [=] AMREX_GPU_DEVICE(amrex::Real z) {
diff --git a/src/problems/NSCBC/dQ_dx_inflow.ipynb b/src/problems/NSCBC/dQ_dx_inflow.ipynb
index 755dd8719..602527f1b 100644
--- a/src/problems/NSCBC/dQ_dx_inflow.ipynb
+++ b/src/problems/NSCBC/dQ_dx_inflow.ipynb
@@ -1,597 +1,1615 @@
 {
-	"cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "from sympy import *\n",
-    "init_printing(use_unicode=True)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Primitive variable system"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# \\partial_t q + F \\partial_x q  + G \\partial_y q + H \\partial_z q = 0\n",
-    "\n",
-    "rho = Symbol('rho', positive=True) # density\n",
-    "p = Symbol('P', positive=True) # pressure\n",
-    "u = Symbol('u') # normal velocity\n",
-    "v = Symbol('v') # transverse velocity\n",
-    "w = Symbol('w') # transverse velocity\n",
-    "s = Symbol('s') # passive scalar\n",
-    "c = Symbol('c') # sound speed\n",
-    "\n",
-    "F = Matrix([\n",
-    "[u,  rho,     0, 0,  0,     0],\n",
-    "[0,  u ,      0, 0,  1/rho, 0],\n",
-    "[0,  0,       u, 0,  0,     0],\n",
-    "[0,  0,       0, u,  0,     0],\n",
-    "[0,  rho*c*c, 0, 0,  u,     0],\n",
-    "[0,  0,       0, 0,  0,     u]])"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}0 & - c \\rho & 0 & 0 & 1 & 0\\\\- c^{2} & 0 & 0 & 0 & 1 & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & 0\\\\0 & c \\rho & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 0   -c⋅ρ  0  0  1  0⎤\n",
-       "⎢                     ⎥\n",
-       "⎢  2                  ⎥\n",
-       "⎢-c    0    0  0  1  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎢ 0    0    1  0  0  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎢ 0    0    0  1  0  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎢ 0   c⋅ρ   0  0  1  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎣ 0    0    0  0  0  1⎦"
-      ]
-     },
-     "execution_count": 3,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the eigenvector matrix S\n",
-    "c = Symbol('c')\n",
-    "rho = Symbol('rho')\n",
-    "lambda_list = [eval for (eval, n, es) in F.left_eigenvects()]\n",
-    "Sinv_list = [[ev for ev in es] for (v, n, es) in F.left_eigenvects()]\n",
-    "Sinv = Matrix([Sinv_list[1][0],  Sinv_list[0][2], Sinv_list[0][0], Sinv_list[0][1], Sinv_list[2][0], Sinv_list[0][3]])\n",
-    "Sinv"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}- c + u & 0 & 0 & 0 & 0 & 0\\\\0 & u & 0 & 0 & 0 & 0\\\\0 & 0 & u & 0 & 0 & 0\\\\0 & 0 & 0 & u & 0 & 0\\\\0 & 0 & 0 & 0 & c + u & 0\\\\0 & 0 & 0 & 0 & 0 & u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡-c + u  0  0  0    0    0⎤\n",
-       "⎢                         ⎥\n",
-       "⎢  0     u  0  0    0    0⎥\n",
-       "⎢                         ⎥\n",
-       "⎢  0     0  u  0    0    0⎥\n",
-       "⎢                         ⎥\n",
-       "⎢  0     0  0  u    0    0⎥\n",
-       "⎢                         ⎥\n",
-       "⎢  0     0  0  0  c + u  0⎥\n",
-       "⎢                         ⎥\n",
-       "⎣  0     0  0  0    0    u⎦"
-      ]
-     },
-     "execution_count": 4,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute eigenvalue matrix \\lambda\n",
-    "lambda_waves = diag(lambda_list[1], lambda_list[0], lambda_list[0], lambda_list[0], lambda_list[2], lambda_list[0])\n",
-    "lambda_waves"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2 c^{2}} & - \\frac{1}{c^{2}} & 0 & 0 & \\frac{1}{2 c^{2}} & 0\\\\- \\frac{1}{2 c \\rho} & 0 & 0 & 0 & \\frac{1}{2 c \\rho} & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & 0\\\\\\frac{1}{2} & 0 & 0 & 0 & \\frac{1}{2} & 0\\\\0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 1     -1          1      ⎤\n",
-       "⎢────   ───  0  0  ────   0⎥\n",
-       "⎢   2     2           2    ⎥\n",
-       "⎢2⋅c     c         2⋅c     ⎥\n",
-       "⎢                          ⎥\n",
-       "⎢ -1                 1     ⎥\n",
-       "⎢─────   0   0  0  ─────  0⎥\n",
-       "⎢2⋅c⋅ρ             2⋅c⋅ρ   ⎥\n",
-       "⎢                          ⎥\n",
-       "⎢  0     0   1  0    0    0⎥\n",
-       "⎢                          ⎥\n",
-       "⎢  0     0   0  1    0    0⎥\n",
-       "⎢                          ⎥\n",
-       "⎢ 1/2    0   0  0   1/2   0⎥\n",
-       "⎢                          ⎥\n",
-       "⎣  0     0   0  0    0    1⎦"
-      ]
-     },
-     "execution_count": 5,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute S\n",
-    "S = Sinv.inv()\n",
-    "S"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\u \\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\\\left(c + u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡(c - u)⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n",
-       "⎢                           ⎥\n",
-       "⎢   ⎛   2                ⎞  ⎥\n",
-       "⎢ u⋅⎝- c ⋅drho_dx + dP_dx⎠  ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢          dv_dx⋅u          ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢          dw_dx⋅u          ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢(c + u)⋅(c⋅du_dx⋅ρ + dP_dx)⎥\n",
-       "⎢                           ⎥\n",
-       "⎣          ds_dx⋅u          ⎦"
-      ]
-     },
-     "execution_count": 6,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the \\mathcal{L} matrix\n",
-    "\n",
-    "#drho_dx = Symbol(r'\\frac{d\\rho}{dx}')\n",
-    "#du_dx = Symbol(r'\\frac{du}{dx}')\n",
-    "#dv_dx = Symbol(r'\\frac{dv}{dx}')\n",
-    "#dw_dx = Symbol(r'\\frac{dw}{dx}')\n",
-    "#dp_dx = Symbol(r'\\frac{dP}{dx}')\n",
-    "#ds_dx = Symbol(r'\\frac{ds}{dx}')\n",
-    "\n",
-    "drho_dx = Symbol('drho_dx')\n",
-    "du_dx = Symbol('du_dx')\n",
-    "dv_dx = Symbol('dv_dx')\n",
-    "dw_dx = Symbol('dw_dx')\n",
-    "dp_dx = Symbol('dP_dx')\n",
-    "ds_dx = Symbol('ds_dx')\n",
-    "\n",
-    "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
-    "L = simplify(lambda_waves * Sinv * dQ_dx_def)\n",
-    "L"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### x1 lower inflow (with relaxation term)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 7,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\\\frac{R c \\eta_{2} \\rho \\left(t - t_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{3} \\left(v - v_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{4} \\left(w - w_{t}\\right)}{L_{x}}\\\\\\frac{c^{2} \\eta_{5} \\rho \\left(1 - M^{2}\\right) \\left(u - u_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{6} \\left(s - s_{t}\\right)}{L_{x}}\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡(c - u)⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n",
-       "⎢                           ⎥\n",
-       "⎢     R⋅c⋅η₂⋅ρ⋅(t - tₜ)     ⎥\n",
-       "⎢     ─────────────────     ⎥\n",
-       "⎢             Lₓ            ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢       c⋅η₃⋅(v - vₜ)       ⎥\n",
-       "⎢       ─────────────       ⎥\n",
-       "⎢             Lₓ            ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢       c⋅η₄⋅(w - wₜ)       ⎥\n",
-       "⎢       ─────────────       ⎥\n",
-       "⎢             Lₓ            ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢  2      ⎛     2⎞          ⎥\n",
-       "⎢ c ⋅η₅⋅ρ⋅⎝1 - M ⎠⋅(u - uₜ) ⎥\n",
-       "⎢ ───────────────────────── ⎥\n",
-       "⎢             Lₓ            ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢       c⋅η₆⋅(s - sₜ)       ⎥\n",
-       "⎢       ─────────────       ⎥\n",
-       "⎣             Lₓ            ⎦"
-      ]
-     },
-     "execution_count": 7,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "eta_2 = Symbol('eta_2')\n",
-    "eta_3 = Symbol('eta_3')\n",
-    "eta_4 = Symbol('eta_4')\n",
-    "eta_5 = Symbol('eta_5')\n",
-    "eta_6 = Symbol('eta_6')\n",
-    "Lx = Symbol('L_x')\n",
-    "M = Symbol('M')\n",
-    "u_t = Symbol('u_t')\n",
-    "v = Symbol('v')\n",
-    "v_t = Symbol('v_t')\n",
-    "w = Symbol('w')\n",
-    "w_t = Symbol('w_t')\n",
-    "s = Symbol('s')\n",
-    "s_t = Symbol('s_t')\n",
-    "t = Symbol('t') # gas temperature\n",
-    "t_t = Symbol('t_t')\n",
-    "R = Symbol('R') # gas constant (kB/mu)\n",
-    "L_inflow_x1_lower = Matrix([L[0],\n",
-    "                            eta_2*(c/Lx)*(rho*R)*(t-t_t),\n",
-    "                            eta_3*(c/Lx)*(v-v_t),\n",
-    "                            eta_4*(c/Lx)*(w-w_t),\n",
-    "                            eta_5*(rho*c**2 * (1 - M**2) / Lx)*(u - u_t),\n",
-    "                            eta_6*(c/Lx)*(s-s_t)])\n",
-    "L_inflow_x1_lower"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 8,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\frac{L_{x} u \\left(c + u\\right) \\left(- c du_{dx} \\rho + dP_{dx}\\right) - 2 R c \\eta_{2} \\rho \\left(c + u\\right) \\left(t - t_{t}\\right) - c^{2} \\eta_{5} \\rho u \\left(M^{2} - 1\\right) \\left(u - u_{t}\\right)}{2 L_{x} c^{2} u \\left(c + u\\right)}\\\\\\frac{L_{x} \\left(c + u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right) - c^{2} \\eta_{5} \\rho \\left(M^{2} - 1\\right) \\left(u - u_{t}\\right)}{2 L_{x} c \\rho \\left(c + u\\right)}\\\\\\frac{c \\eta_{3} \\left(v - v_{t}\\right)}{L_{x} u}\\\\\\frac{c \\eta_{4} \\left(w - w_{t}\\right)}{L_{x} u}\\\\\\frac{L_{x} \\left(c + u\\right) \\left(- c du_{dx} \\rho + dP_{dx}\\right) - c^{2} \\eta_{5} \\rho \\left(M^{2} - 1\\right) \\left(u - u_{t}\\right)}{2 L_{x} \\left(c + u\\right)}\\\\\\frac{c \\eta_{6} \\left(s - s_{t}\\right)}{L_{x} u}\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                                                                   2        ⎛\n",
-       "⎢Lₓ⋅u⋅(c + u)⋅(-c⋅du_dx⋅ρ + dP_dx) - 2⋅R⋅c⋅η₂⋅ρ⋅(c + u)⋅(t - tₜ) - c ⋅η₅⋅ρ⋅u⋅⎝\n",
-       "⎢─────────────────────────────────────────────────────────────────────────────\n",
-       "⎢                                            2                                \n",
-       "⎢                                      2⋅Lₓ⋅c ⋅u⋅(c + u)                      \n",
-       "⎢                                                                             \n",
-       "⎢                                                   2      ⎛ 2    ⎞           \n",
-       "⎢                 Lₓ⋅(c + u)⋅(c⋅du_dx⋅ρ - dP_dx) - c ⋅η₅⋅ρ⋅⎝M  - 1⎠⋅(u - uₜ)  \n",
-       "⎢                 ──────────────────────────────────────────────────────────  \n",
-       "⎢                                      2⋅Lₓ⋅c⋅ρ⋅(c + u)                       \n",
-       "⎢                                                                             \n",
-       "⎢                                        c⋅η₃⋅(v - vₜ)                        \n",
-       "⎢                                        ─────────────                        \n",
-       "⎢                                             Lₓ⋅u                            \n",
-       "⎢                                                                             \n",
-       "⎢                                        c⋅η₄⋅(w - wₜ)                        \n",
-       "⎢                                        ─────────────                        \n",
-       "⎢                                             Lₓ⋅u                            \n",
-       "⎢                                                                             \n",
-       "⎢                                                    2      ⎛ 2    ⎞          \n",
-       "⎢                 Lₓ⋅(c + u)⋅(-c⋅du_dx⋅ρ + dP_dx) - c ⋅η₅⋅ρ⋅⎝M  - 1⎠⋅(u - uₜ) \n",
-       "⎢                 ─────────────────────────────────────────────────────────── \n",
-       "⎢                                         2⋅Lₓ⋅(c + u)                        \n",
-       "⎢                                                                             \n",
-       "⎢                                        c⋅η₆⋅(s - sₜ)                        \n",
-       "⎢                                        ─────────────                        \n",
-       "⎣                                             Lₓ⋅u                            \n",
-       "\n",
-       " 2    ⎞         ⎤\n",
-       "M  - 1⎠⋅(u - uₜ)⎥\n",
-       "────────────────⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎥\n",
-       "                ⎦"
-      ]
-     },
-     "execution_count": 8,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "dQ_dx_inflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_inflow_x1_lower)\n",
-    "dQ_dx_inflow_x1_lower"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "dQ_dx[0] = (1.0/2.0)*(L_x*u*(c + u)*(-c*du_dx*rho + dP_dx) - 2*R*c*eta_2*rho*(c + u)*(t - t_t) - std::pow(c, 2)*eta_5*rho*u*(std::pow(M, 2) - 1)*(u - u_t))/(L_x*std::pow(c, 2)*u*(c + u));\n",
-      "dQ_dx[1] = (1.0/2.0)*(L_x*(c + u)*(c*du_dx*rho - dP_dx) - std::pow(c, 2)*eta_5*rho*(std::pow(M, 2) - 1)*(u - u_t))/(L_x*c*rho*(c + u));\n",
-      "dQ_dx[2] = c*eta_3*(v - v_t)/(L_x*u);\n",
-      "dQ_dx[3] = c*eta_4*(w - w_t)/(L_x*u);\n",
-      "dQ_dx[4] = (1.0/2.0)*(L_x*(c + u)*(-c*du_dx*rho + dP_dx) - std::pow(c, 2)*eta_5*rho*(std::pow(M, 2) - 1)*(u - u_t))/(L_x*(c + u));\n",
-      "dQ_dx[5] = c*eta_6*(s - s_t)/(L_x*u);\n"
-     ]
-    }
-   ],
-   "source": [
-    "drho_dx = Symbol('drho_dx')\n",
-    "du_dx = Symbol('du_dx')\n",
-    "dv_dx = Symbol('dv_dx')\n",
-    "dw_dx = Symbol('dw_dx')\n",
-    "dp_dx = Symbol('dP_dx')\n",
-    "ds_dx = Symbol('ds_dx')\n",
-    "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
-    "L = simplify(lambda_waves * Sinv * dQ_dx_def)\n",
-    "\n",
-    "L_inflow_x1_lower = Matrix([L[0],\n",
-    "                            eta_2*(c/Lx)*(rho*R)*(t-t_t),\n",
-    "                            eta_3*(c/Lx)*(v-v_t),\n",
-    "                            eta_4*(c/Lx)*(w-w_t),\n",
-    "                            eta_5*(rho*c**2 * (1 - M**2) / Lx)*(u - u_t),\n",
-    "                            eta_6*(c/Lx)*(s-s_t)])\n",
-    "\n",
-    "dQ_dx_inflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_inflow_x1_lower)\n",
-    "print(cxxcode(dQ_dx_inflow_x1_lower, assign_to='dQ_dx'))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### x1 lower inflow (with relaxation term; zero normal velocity)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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ziMZM2Vm+4snWM5r2ULcxqj2d7qozdX+JAKMxJapj3QcS8di2XgMrDIhCjehAC1DQUo3dLZu3wV80oSMPz0m00IVlET3WZfmqHDMw5KOlvbaPCqIuXRo5It9+8FpV/MhW0/nbr3W+hmg7pnoHLrTjrSUnzms/GwdAvV1IPj1oVexV09nyP9YTmge5mxdhCS+2dNHsJ/lznUXI8lUZtDGsH0XHjgHq2xkn7ijcuBDkna2483fUjmZV1YmALtlBSsMk8GaBws6+NaaFBQBayd3iAiBSDg1OGaHJYejiMlKZx8T18GXwmZlBdK2DO7ZdJ/mmnWPK31CeP1xdrtzLBfq+w2m7OhatxXRL52M3vtDDDllDmykM+DE5nFZX8KzOmCOqz6FuY+yTKKbCpbrU7YxFU7tAAsBXv3kx1VGqR9css9vbGFNtDeMuFriALxTE0Hflx5bOmRND2S1Fv9vbGPsEchjgCkRnuc1R5fbJ+KzpqfopztnxZ63blMKPBNzdd8aUjry0vJf8cXZpfX2o9h5G456rVzTt3qhs7GW2iNkAqLc1SghLuwrciRIWcOttjRNlOCZ7NRXGSK2dh50pv7nRTq4xc0ugAnceiWIu+DMP87CsXLokUIHbJZ2CNJkKbB3jqsb9IIdV/gJct/PCCaTqhksAbcupsiOcBRje+nVzoBzMxhHAfaiHc6cP9FQ3XAJJ+1ZA5jwDoK5uPgkgT3NG+no+nhfLCWGmto85pHOLVARgzAmOOtbbGhHIDK7auBOEaAEJh4Z9q3jWdOttjRNk25e1atw+CWXSBU6OPvLzHhynxfA5dINmRcNiQhinNHfqDO1sVh8U1zrPa8mrVyCBCtwCIaVIBDx+TRH/oiJFehKtu8MLMBvAKq516DuZuUYmJVCBmxTL7JHxbY1o5cbv2WYv8eAMK3AHdLC0JLZr0Q2NIVvlC3/5Gyat9h5o/dXKXLKg+nFWKF2mdpHudr1W9T/UzY0VuAXAVafz0bWnGxpzrTrMzY0VuLkubsbzy16z8N2M3ldIA5Bluxs7EPdV+ai2h7Fx1RmAa7UbGlXejcpj2Su8nRHNzI8oWR7j/f/18G+hRm0HL1SGAa/qlbs8REnbd2hcdxGI87df66iG6uBVb2i0oEzdzvhEVXtI9URzJw/AmjBxQ9yCZez55kZ/IQjAfWcF6vwh8j07rTqYKbzrhka0Ioe9Qw2DNmwdKhIN15OSFjqAF9M+UFzrdkbFsU4bHm98pfAobat8o8pQ/dmx6zJrGFCjBpPyndv5C0GOYCqg5fxykzqNjvnNSVhhwNR1Q6MhFR2rBqkOBcjw9E60Bozy2TkLBwTmQwM0AS0DCHOmz2FmZG9qVOZsGZYx6V33hMWDsK8+m0w/AnABVpdWY7E/3F6lY1NnC4gL6RQ0Dt6UkXKA0QBVYIOGE2Ehbw94xTOY/DZwilkmbkgZbENz4OfwNzdiKuzdAYiGphRIjGaTTzxg8hpY4cZ0rjSAkV2ftXkb/EXvHMB0GowB4p3y5a5E8jSFL8VlqEyzBS0frR3OBGFRtKVLI4e0m30/gsYdfUOjOhct6f7Pblcn5W5oRENzVT8fPAAMwDBoAIaf7vU+xRWXYdtDPbocAzd1DLMrz+bSdg9cdRZTebIjlMa07aduhWPt+kDpaGSmWBzaCCByhX5ofgBGymiYEqJp8Ff67G5gGZhBb6iE8rUO8SiOgUpa2DaidueudlfjaRX2HQsbOlAP06s7dpjkLhoACsBzJkMy3xkijUmkeubMFDcAz1C1eYvcvcYdKI7WDY3kV0czvQJgYyMSFzvRGDNA/q2lj0nOHla9umYfwMyW7+61LYK+KOCq05ImRSnilH8PNzTmmnOomxt3DVwBqd7QmINpIl7ySsSaGWd3FwbuHbi7E3gSOTVysAQu7eNssIBqhm1KAI3rDHq3kL7Nmm60Vpp+b1Q1bGd+HcGif72tUUJYyLG6Y75T0Lgs8bBtyZpmdQMlIOAiTLMBoXf+wV92ZWIg60relgBKwmyxV1OhLZwxMexGAeDqVpJABe48gkYThMcZ5+FauWQlUIGbFU1ZgkwDtpFxVeN+kMMqfytwp4sZbZs9XTadfeWQkkAFbkoqw+KS9q00MWcbAHV1C0jgegGel8YScKa2kl8o/hZhWHOC87r1tkYEMoOrGneCEC0g4dCwbxXPmm69rXGCbPuyVo3bJ6FMusDJGd6nNpmTY7xy3hXNygcbJoRxSnPHJtHO9bZGK5cpXgXuSOkJjJ3HIEO2gNzSA2ZzGF3h1kHvME9975ZABW63fOZKrbc1ziVJywfgfmTfnT9zEZWdtKv/+XyXNESHXbzrg94Lt+FTJz+A+84GnO/SNudLKDeqFHbkQz3m1wyKa3wYba7ShRVSO7CZAe2mgVvQBwy+x3qWOLPxhxPn3lYV+OnJj1Yo2IqH2Galo9WWT2ib65gN+519oDYYRSKfnwot5vYG3M8DSaB1+a3Yrp06mBmEXxY3fkG84Ub19oHagrZlpYW2LeJ2BVwJIpxG+cVqke24iOTmY0o7zFG9+Vgux2lAH9Am2raI2+WqgoSHUPjhovk59iKSWY/pE7UjtfN2Ujwa64ketFeo6SbVTrxYZ2ZnDxMFOSbtUcUjZ6Z81qG5a8LL26Zl+0DpXAr4px5oZp8Zd6VxJbyThMAoNgLTOx80u3WqP8AJZxHfFqUBLkADeGedcsUbAPJ9gE39SE/LiYYP4Qd6uIeMgROCtrQPaBsDb3a3K+BKeL9KAoCVkcwvfJOaanYpLccweUCH4gCKHjRhEtgzVAkN/koPA6ThVK4bLPiND2ClDekDPtT8DmKjkImBzZoKEhCjOr5hvHe6VD60FHQImE7nI46lM5Zpim1i0dJpTKfUAa10qzijdeSjjehwfxu53k+K54u7uAxlYXuY6fhcjvYgn9hhvjDVcz1r46NRcb19EDDjDjVkNbvbpMaVcGgwlyyjcbgLCwD1OtG7KRWBAwiOFtIBCP9Gvukk6FLvUQE/K/xKdNSBU16ET+SV17qNXPEAeWgnweutnlWd6oocfpVvtLkNmzronTZgHhhNLN+bCCMqSduMzEfk7cxy1Zl6hkQJCsBxp5fbWEC4XxZW5YHoXutB6F23kGN3/a4yMDcAeQM8iscceSPfdRp1AOwAjTJSt5HT4QyyhlMeNH1Oq8LrrpFhnYABpy2KNiKvk20f7aS9IQ3JY5yR25iMfXmu+wjOkA6o/HQrIdKxDkCd1bECP8nHrgKQxilMx9zJ9yDRe9dlItjOof3MlH6y+Q0PvT9VVAhIzImGPUgeOQDADDKrU/m0CXOqz/FhBYBCxz9UcSfWGGzUnQ81YyJYQmSYao9NLvIWG5gA9yNbBecX1WhBovviHQt6aHEALbTNWtpDHUc50DXusbXxTG+h9mQg0bGhg2cIXMLYwZgRZqDJR3MzAHI3hL9VGvUY7GwZc3z4MKj4H26xjAiHMhxcR2WgbbRxLvepY4Sp8M4GnO/SzuXT6Q27SEIt0SymvqIlL9OdAY9tRKw9ECY2L+YI++rh9iQddlLcnfXhR9xXhCNnBphoSaeTCJv80Cke+xgfrRcDnyToG20lckmnelBeOAMgJwawMxGoL9ocP5QhUUMdGneqEgrL/MMFrt3Lhny0G+u0rkMBBB9HpY58BjBBBsAUTv1oQANMxTNl/ml9eWb5hvMQaEtoWJFganX0ChoHkKknIKBzACYDwHe20qh7nE9R3kHLNN1yyks7qDN1x042qyzy47a18uYilNfNAJhNLLcxcJlZsOfNwLI0DHRo+J9tU7QuKxBeHnqfzW0OuBIUIAhBNqixyk9nuA+7k8JoDzohHPn847yPA8YhuNA+TPkpDemz2PSQJnx3dIDuDQHRP9Pj7EqX/kov3hZ3kfiipb5TQBOyM+/iCegbwFccbQeoxqVoXNoIP1YYI1iks1ylow8V68ETtOordZAzEV4oPly1AOge+EGeMa9G26gsNHGLp+JdupmaxxSw1Txqm5kx5bfaPUedN6dx52hUxIOpvqHVJMyUdjxZYYdmRMRqWFD8SmYPTABmmNGzzLBarUbNbFH8bTK0VocHrsBTDAgLND9tDhXmGHqVyZowmy3YsaE5M4bdJvLQFlWE9hTLfmjFL8FUGCqTc9DzQRourZ2jDnOWSVsWAy0VrcCds7tG8pJm4gOJJTO++nftbBtoy6Kzx+FNhb2gwHZ044t/L3UP66l2rNKGWYCryt5X5X/Wg1/quAS5scan8Fn+GUlphSvddAmoj7u22osLmAu4THVDjrslKzhXo5LMa+ShJFBt3EN15+U05t7793V2XrK7NYvciD9f2I/1sH5c/7mJhDDV3Xv58iW7Nvxa4DMJedEvwamV3Wt+yZXVAn6CHm4z77U5Z6u35McOJD/avFdNhXW6gU2NRbY+16n+9kqpwF2nTzAXph7KXqemOymlAnfhjtK05g7QVI07o6wrcGcUZoYV2jY+VpkhrdGlEqjALZXUeLqkfStNfF8PoK5uhASuR+SpWYZJAHCmDpxwDvgWVtac4AA7RzDdXRL86iKVjywX76rGXRACFpCU0LBvFc+aLsf+2HHEfaF3fh3BLyL44SLvLk2v1cUSqBo3lshMYYGPtdunlh2npXjlLAealQ82f+7XAlVR5vdlZvVBcbP+bAfmR3IVuAv1poDX+n1XrijR1n9ukhNOJr4CNyOYlaPrPzcZKPAK3IECC8mlKbFV+Zn3pK1y5ecXELO6QIvPyncrzOrH2cieEDDYN9/s+qzq962eI/0cqNFTFbgNcZQFBAg+svj6b6wWlOVelYprTxf7pe2qLYkKq8CNBFIYBAyb12Z2YN3YgVbYtH2QHdbGtZrGLeazS8XUyfQ++dJn8QAMyc0BW667m+utwsk7HIbAYyJPZoUbPZPrMaTOS9MeUuOqowFO42JoxTG947gcBG0JmFuXPkPQ5ZTnodKTmwNKgy+XQbOB8EaPW8fV6zg3A89fVLJfMx5Xi+3lOpzGtR2duhj6gcT/Wo+5Dkl0oQYC1G9d9yiN1QIuozOaU+Hwzi+A62ldHuuT9oOtw2v5c6wWdPJUGaRz+V5yBlAagwyaQ7nDAVe980SPB4w6lI7zvyZWGO3jr2RSmF0sVgeMFpUPsPk1CGAA0NCGwCXO0MqPHbtd/NoZzctNh5w3MEtl8imn5EMpvpMgy1P8cJgBzjQxEdGf3CCLyPYVPCJwAVbXuipbruF2Kh0frg4ALgBHPJo7nmbhTRkNJzq0GvQf23e2btHc5p4BxTF4Yl6KyrsCnmwro2lzF0fD3NSLlyO5qyM1xrYFgDSmRgHAaDoLBGxbr4FFC5jc+QAAycOt4oAZv7F6YPM2+IsOBw8GxUk0gBtTJBwQJA11nTxVjhsUaOnQ9AnLoa5dGjmk3c37ETUuZkLuYmg60XR20ENo1oZ96ECArweblWk/1OJo1viSOgYHgwJTA8evecMB8iF22N9OnuLPILvrYcnAbLSvh34XyYcDrjoTgCU7SmloQK8FFXb2rQGlwti6bOECwBAQb6PeBFCU4U0O0Yd2cEQ+LljAk0HH6sVJtM9ieoUBNmnhoCNq9+5q9y2Y1gDf8QEbNDYalhNbmAlcFRWC+KQw4AfcKZMhYLX4q9HoqodZKUmU5gZYImnfUYfTuAO7I3Xpc1Zjh7wFFuxKAH6rpwHskG7Jd5WbravSADNbvofTtsj0ooGrTk2aFKVgU363ZFaaZU061pHPMqDWaOShgKuOOst9Uip3jb4aXEaqXoqb5bbEwZWZOcPRgHuITpm5jw/J7tI/zg7ZqZfQqENp3HN0mKbeG5WLrcwuGZsA9TZGCWFpV4E7UcICLtutrAffyPdnJCayrdl7JFBNhR4BFSazO+U3NgrzVLIJEqjAnSC8ICvmgjnvEMTV1wUlUIE7UbjWTIBL1bgTZTkkewXuEGmladG2nHE45A5Vusnnj63And4HSftWQOYsA6CubgEJhKsKv0vQcREc66tfyrFUmmHAmeEkYgcAAACiSURBVNo6fqH4W0glQ1YdOKtbb2NEIAVOMuMM8cMcKcDNHtSwabm8Fx9vAYkcGvat4lnTbd3GaOk5mMOdDJzcqi4vgU751H8XlRdcZ4qA525jRJu6w+mcf0WzEte6MMTmOcl39CKrbpQE+D9n9VlWBvqXXM+Rsfyf+Pdc9v1Zlft4uVeNO2q4D8skDcsvhVnnRSN/ogf77Y3ip/60R2wu0/0XjKK+EieIZeEAAAAASUVORK5CYII=",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}c \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\\\frac{R c \\eta_{2} \\rho \\left(t - t_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{3} \\left(v - v_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{4} \\left(w - w_{t}\\right)}{L_{x}}\\\\- \\frac{c^{2} \\eta_{5} \\rho u_{t} \\left(1 - M^{2}\\right)}{L_{x}}\\\\\\frac{c \\eta_{6} \\left(s - s_{t}\\right)}{L_{x}}\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡c⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n",
-       "⎢                     ⎥\n",
-       "⎢  R⋅c⋅η₂⋅ρ⋅(t - tₜ)  ⎥\n",
-       "⎢  ─────────────────  ⎥\n",
-       "⎢          Lₓ         ⎥\n",
-       "⎢                     ⎥\n",
-       "⎢    c⋅η₃⋅(v - vₜ)    ⎥\n",
-       "⎢    ─────────────    ⎥\n",
-       "⎢          Lₓ         ⎥\n",
-       "⎢                     ⎥\n",
-       "⎢    c⋅η₄⋅(w - wₜ)    ⎥\n",
-       "⎢    ─────────────    ⎥\n",
-       "⎢          Lₓ         ⎥\n",
-       "⎢                     ⎥\n",
-       "⎢  2         ⎛     2⎞ ⎥\n",
-       "⎢-c ⋅η₅⋅ρ⋅uₜ⋅⎝1 - M ⎠ ⎥\n",
-       "⎢─────────────────────⎥\n",
-       "⎢          Lₓ         ⎥\n",
-       "⎢                     ⎥\n",
-       "⎢    c⋅η₆⋅(s - sₜ)    ⎥\n",
-       "⎢    ─────────────    ⎥\n",
-       "⎣          Lₓ         ⎦"
-      ]
-     },
-     "execution_count": 10,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "L_inflow_x1_lower_zerou = L_inflow_x1_lower.subs(u, 0)\n",
-    "L_inflow_x1_lower_zerou"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}- \\frac{1}{c} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & \\frac{1}{c} & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡-1                ⎤\n",
-       "⎢───  0  0  0  0  0⎥\n",
-       "⎢ c                ⎥\n",
-       "⎢                  ⎥\n",
-       "⎢ 0   0  0  0  0  0⎥\n",
-       "⎢                  ⎥\n",
-       "⎢ 0   0  0  0  0  0⎥\n",
-       "⎢                  ⎥\n",
-       "⎢ 0   0  0  0  0  0⎥\n",
-       "⎢                  ⎥\n",
-       "⎢              1   ⎥\n",
-       "⎢ 0   0  0  0  ─  0⎥\n",
-       "⎢              c   ⎥\n",
-       "⎢                  ⎥\n",
-       "⎣ 0   0  0  0  0  0⎦"
-      ]
-     },
-     "execution_count": 11,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "def pinv(x): # Moore-Penrose pseudoinverse of a diagonal matrix\n",
-    "    if x != 0:\n",
-    "        return x**-1\n",
-    "    else:\n",
-    "        return 0\n",
-    "ll = [pinv(l.subs(u,0)) for l in lambda_list]\n",
-    "linv_zerou = diag(ll[1], ll[0], ll[0], ll[0], ll[2], ll[0])\n",
-    "linv_zerou"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 12,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\frac{L_{x} \\left(- c du_{dx} \\rho + dP_{dx}\\right) + c \\eta_{5} \\rho u_{t} \\left(M^{2} - 1\\right)}{2 L_{x} c^{2}}\\\\\\frac{L_{x} \\left(c du_{dx} \\rho - dP_{dx}\\right) + c \\eta_{5} \\rho u_{t} \\left(M^{2} - 1\\right)}{2 L_{x} c \\rho}\\\\0\\\\0\\\\\\frac{L_{x} \\left(- c du_{dx} \\rho + dP_{dx}\\right) + c \\eta_{5} \\rho u_{t} \\left(M^{2} - 1\\right)}{2 L_{x}}\\\\0\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                                    ⎛ 2    ⎞⎤\n",
-       "⎢Lₓ⋅(-c⋅du_dx⋅ρ + dP_dx) + c⋅η₅⋅ρ⋅uₜ⋅⎝M  - 1⎠⎥\n",
-       "⎢────────────────────────────────────────────⎥\n",
-       "⎢                        2                   ⎥\n",
-       "⎢                  2⋅Lₓ⋅c                    ⎥\n",
-       "⎢                                            ⎥\n",
-       "⎢                                   ⎛ 2    ⎞ ⎥\n",
-       "⎢Lₓ⋅(c⋅du_dx⋅ρ - dP_dx) + c⋅η₅⋅ρ⋅uₜ⋅⎝M  - 1⎠ ⎥\n",
-       "⎢─────────────────────────────────────────── ⎥\n",
-       "⎢                  2⋅Lₓ⋅c⋅ρ                  ⎥\n",
-       "⎢                                            ⎥\n",
-       "⎢                     0                      ⎥\n",
-       "⎢                                            ⎥\n",
-       "⎢                     0                      ⎥\n",
-       "⎢                                            ⎥\n",
-       "⎢                                    ⎛ 2    ⎞⎥\n",
-       "⎢Lₓ⋅(-c⋅du_dx⋅ρ + dP_dx) + c⋅η₅⋅ρ⋅uₜ⋅⎝M  - 1⎠⎥\n",
-       "⎢────────────────────────────────────────────⎥\n",
-       "⎢                    2⋅Lₓ                    ⎥\n",
-       "⎢                                            ⎥\n",
-       "⎣                     0                      ⎦"
-      ]
-     },
-     "execution_count": 12,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "dQ_dx_inflow_x1_lower_zerou = simplify(S * linv_zerou * L_inflow_x1_lower_zerou)\n",
-    "dQ_dx_inflow_x1_lower_zerou"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 13,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "dQ_dx[0] = (1.0/2.0)*(L_x*(-c*du_dx*rho + dP_dx) + c*eta_5*rho*u_t*(std::pow(M, 2) - 1))/(L_x*std::pow(c, 2));\n",
-      "dQ_dx[1] = (1.0/2.0)*(L_x*(c*du_dx*rho - dP_dx) + c*eta_5*rho*u_t*(std::pow(M, 2) - 1))/(L_x*c*rho);\n",
-      "dQ_dx[2] = 0;\n",
-      "dQ_dx[3] = 0;\n",
-      "dQ_dx[4] = (1.0/2.0)*(L_x*(-c*du_dx*rho + dP_dx) + c*eta_5*rho*u_t*(std::pow(M, 2) - 1))/L_x;\n",
-      "dQ_dx[5] = 0;\n"
-     ]
-    }
-   ],
-   "source": [
-    "print(cxxcode(dQ_dx_inflow_x1_lower_zerou, assign_to='dQ_dx'))"
-   ]
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.11.5"
-  },
-  "orig_nbformat": 4
- },
- "nbformat": 4,
- "nbformat_minor": 2
+	"cells" : [
+		{
+			"cell_type" : "code",
+			"execution_count" : 1,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [ "from sympy import *\n", "init_printing(use_unicode=True)" ]
+		},
+		{"cell_type" : "markdown", "metadata" : {}, "source" : ["## Primitive variable system"]}, {
+			"cell_type" : "code",
+			"execution_count" : 2,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [
+				"# \\partial_t q + F \\partial_x q  + G \\partial_y q + H \\partial_z q = 0\n", "\n",
+				"rho = Symbol('rho', positive=True) # density\n", "p = Symbol('P', positive=True) # pressure\n",
+				"u = Symbol('u') # normal velocity\n", "v = Symbol('v') # transverse velocity\n", "w = Symbol('w') # transverse velocity\n",
+				"s = Symbol('s') # passive scalar\n", "c = Symbol('c') # sound speed\n", "\n", "F = Matrix([\n",
+				"[u,  rho,     0, 0,  0,     0],\n", "[0,  u ,      0, 0,  1/rho, 0],\n", "[0,  0,       u, 0,  0,     0],\n",
+				"[0,  0,       0, u,  0,     0],\n", "[0,  rho*c*c, 0, 0,  u,     0],\n", "[0,  0,       0, 0,  0,     u]])"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 3,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "TYkIIHEGgyprzCD1lCQEhMBMBGedMoFRMCLRGQMbZGnHREwIzEZBxzgRKxYRAawSGP6X8jp2iMf03SOPHsgpCQAgURgC2xf/"
+					    "3ejVVLY2T51yfTxQIewZ2Qh4lC4EtIWAO8JI8/R+r53RWtVJV8QAAAABJRU5ErkJggg==",
+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}0 & - c \\rho & 0 & 0 & 1 & 0\\\\- c^{2} & 0 & 0 & 0 & 1 & "
+							"0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & 0\\\\0 & c \\rho & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & "
+							"0 & 1\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡ 0   -c⋅ρ  0  0  1  0⎤\n", "⎢                     ⎥\n", "⎢  2                  ⎥\n",
+						"⎢-c    0    0  0  1  0⎥\n", "⎢                     ⎥\n", "⎢ 0    0    1  0  0  0⎥\n",
+						"⎢                     ⎥\n", "⎢ 0    0    0  1  0  0⎥\n", "⎢                     ⎥\n",
+						"⎢ 0   c⋅ρ   0  0  1  0⎥\n", "⎢                     ⎥\n", "⎣ 0    0    0  0  0  1⎦"
+					]
+				},
+				"execution_count" : 3,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the eigenvector matrix S\n", "c = Symbol('c')\n", "rho = Symbol('rho')\n",
+				"lambda_list = [eval for (eval, n, es) in F.left_eigenvects()]\n",
+				"Sinv_list = [[ev for ev in es] for (v, n, es) in F.left_eigenvects()]\n",
+				"Sinv = Matrix([Sinv_list[1][0],  Sinv_list[0][2], Sinv_list[0][0], Sinv_list[0][1], Sinv_list[2][0], Sinv_list[0][3]])\n",
+				"Sinv"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 4,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "kUMbmTpJB051Agb4SI8vXEXiOoMNyBr+c2QIOhu9IkRMZqyh/"
+					    "wsY5ol++"
+					    "QJ7OhXEoAJWGOmPDmlSWJmenUQXe643TWz5NE0EHiybSAo6EAWs6Dl5nq4KNhXEonYFNgg1ZNJbCdXjj9JZPeiPoYGHmVnDw8ffqYGPRHIpJsCGLVl"
+					    "K4Dm+c3vJJbwQdLMzcCo4LkHGTEfLgYGOhHAq6XQr0pUBfHMc/"
+					    "zxp22M2G2uxgOgG4OFn74GBjYd5D6bWYK+"
+					    "QV6KtHSKFsBJ4j6GBBbws4TIKNKdCXRXNSHWJgxwygN6NAXzu2v6CLgWIMhJpDKYZSFYsBMVCFATmUKjRLiBjYBwNyKPuws1CKgSoMyKFUoVlCxMA+"
+					    "GOg/Nv6I2doc9VuU8Q06JTEgBsTAAf7gI2i4uI8KOhS+0PLingMGL7vcc4yKxYAY2A8D6QPyo4j/D04MQISuHGg1AAAAAElFTkSuQmCC",
+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}- c + u & 0 & 0 & 0 & 0 & 0\\\\0 & u & 0 & 0 & 0 & 0\\\\0 & 0 & u & 0 & 0 "
+					     "& 0\\\\0 & 0 & 0 & u & 0 & 0\\\\0 & 0 & 0 & 0 & c + u & 0\\\\0 & 0 & 0 & 0 & 0 & u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡-c + u  0  0  0    0    0⎤\n", "⎢                         ⎥\n", "⎢  0     u  0  0    0    0⎥\n",
+						"⎢                         ⎥\n", "⎢  0     0  u  0    0    0⎥\n", "⎢                         ⎥\n",
+						"⎢  0     0  0  u    0    0⎥\n", "⎢                         ⎥\n", "⎢  0     0  0  0  c + u  0⎥\n",
+						"⎢                         ⎥\n", "⎣  0     0  0  0    0    u⎦"
+					]
+				},
+				"execution_count" : 4,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute eigenvalue matrix \\lambda\n",
+				"lambda_waves = diag(lambda_list[1], lambda_list[0], lambda_list[0], lambda_list[0], lambda_list[2], lambda_list[0])\n",
+				"lambda_waves"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 5,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "OJfa+PVlr+YMIM/KOZQrMVA8hDN+C/5xUAkR3aZHjPKvT3s8pOvQk/"
+					    "ToccKXuELHyizprs2Xm35YxtNHjvw42IOpVsMXFPiybBxEgb7Q5wu3G2Crq4W2tHGqy1/piOa9uMaHIq2GED6ZwDNxUPY1GHiPynzLrHdbnP6/"
+					    "GDrwNNCO9p4teXnvNCDHxdxKB0YOLnppxzrvZUj6LG7QN1dJG282vJnGMm8H5dyKBoYAJbN6nEiwb8gv+"
+					    "fWwljfOO6AgTX4sfQYw84tQO4lvvBNTmzyRAoGXDLQqx+rBAaQyTsSHF+"
+					    "4wrG1UXmXDhqg5Rno2Y9FuxI03UAm70x8PPrOfDMTfeRdLCR6Y6B3PxYNDAOZz+"
+					    "AEDAzsSjBxnGFND1ZtlY4PvwyswY9FAwNcgXMW7g37nWeAaDcj9jvQcbBmBrr3Y9HAgADw/pq9KXTvg4E1+LHK4GMf7hFaBAP9MhCBoV/"
+					    "bhmbBwMkMRGA4mbr4YTDQLwOiYwykEf2z9KQhX1DyEBufIej+VqU3vbXxasunrx5L1vEl7MDJu3982nLRnCHRFgNAcjT3Bfa8XckXwPI25Q84vsC+"
+					    "2+"
+					    "RNb2282vJzjugA3z1g5MOKnBrwLbbF15dYYABIVwuv5Jxjbrk3vbXxasvP2dU6PuIHRr457BE2TgP4LqfTVLlYYIDw4kUwphRwkOdNb2282vJzLmUd"
+					    "Xw7/rHLJwHANRFMvqkzjCyzvMXnTWxuvtvycD1rHl8M/q1wkMKBJw9mOuZTeApw7z025N7218WrLzzmWdXw5/EvKRQIDAKWL/"
+					    "tiTlHOCxxLdLJzrTW9tvNrycz5jHV8O/+xyqcAwB9CDOSd1eI43vbXxasvPuaB1fDn823KpwDA1tpAApijMeQ29JW96a+PVlp/"
+					    "zP+v4cvhnl4sEBvTNUhdiqruQ8tIg5Gzw1k/0prc2Xm35OX+yji+Hf0m5SGAYABUvgrFEMUPnetNbG6+2/"
+					    "JzrWMeXwz+rXDIwcFrmBxOorpBnceGVCagnZXnTWxuvtvycka3jy+GfVS4WGNAMc7Xwyiz2ZpzkTW9tvNrycya1jm8CfxoMTWN5E6e8m3X+"
+					    "blbTHLYOPC28UosMb3pr49WWn7O7dXwbBDC2bJg4IYuJz05wHI/"
+					    "PJu0v47A9YfxxdnNzc4kMPtz0cPjhuDyOg4FgYCUM4Prn80zPsD8T60qshNtQMxjogoEIDF2YMZQIBuoyEIGhLp9RWzDQBQPjwcef0bfYV+"
+					    "o58viYaaRgIBjoiAFc1z9DnYtDKjEwcKTy0LoO3c1GPERE5AcDK2MgvWJxUu3/AySyMXnbmOhyAAAAAElFTkSuQmCC",
+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2 c^{2}} & - \\frac{1}{c^{2}} & 0 & 0 & \\frac{1}{2 c^{2}} & "
+					     "0\\\\- \\frac{1}{2 c \\rho} & 0 & 0 & 0 & \\frac{1}{2 c \\rho} & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 "
+					     "& 0\\\\\\frac{1}{2} & 0 & 0 & 0 & \\frac{1}{2} & 0\\\\0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡ 1     -1          1      ⎤\n", "⎢────   ───  0  0  ────   0⎥\n", "⎢   2     2           2    ⎥\n",
+						"⎢2⋅c     c         2⋅c     ⎥\n", "⎢                          ⎥\n", "⎢ -1                 1     ⎥\n",
+						"⎢─────   0   0  0  ─────  0⎥\n", "⎢2⋅c⋅ρ             2⋅c⋅ρ   ⎥\n", "⎢                          ⎥\n",
+						"⎢  0     0   1  0    0    0⎥\n", "⎢                          ⎥\n", "⎢  0     0   0  1    0    0⎥\n",
+						"⎢                          ⎥\n", "⎢ 1/2    0   0  0   1/2   0⎥\n", "⎢                          ⎥\n",
+						"⎣  0     0   0  0    0    1⎦"
+					]
+				},
+				"execution_count" : 5,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "# compute S\n", "S = Sinv.inv()\n", "S" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 6,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - "
+							"dP_{dx}\\right)\\\\u \\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\\\left(c "
+							"+ u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡(c - u)⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n", "⎢                           ⎥\n", "⎢   ⎛   2                ⎞  ⎥\n",
+						"⎢ u⋅⎝- c ⋅drho_dx + dP_dx⎠  ⎥\n", "⎢                           ⎥\n", "⎢          dv_dx⋅u          ⎥\n",
+						"⎢                           ⎥\n", "⎢          dw_dx⋅u          ⎥\n", "⎢                           ⎥\n",
+						"⎢(c + u)⋅(c⋅du_dx⋅ρ + dP_dx)⎥\n", "⎢                           ⎥\n", "⎣          ds_dx⋅u          ⎦"
+					]
+				},
+				"execution_count" : 6,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the \\mathcal{L} matrix\n", "\n", "#drho_dx = Symbol(r'\\frac{d\\rho}{dx}')\n",
+				"#du_dx = Symbol(r'\\frac{du}{dx}')\n", "#dv_dx = Symbol(r'\\frac{dv}{dx}')\n", "#dw_dx = Symbol(r'\\frac{dw}{dx}')\n",
+				"#dp_dx = Symbol(r'\\frac{dP}{dx}')\n", "#ds_dx = Symbol(r'\\frac{ds}{dx}')\n", "\n", "drho_dx = Symbol('drho_dx')\n",
+				"du_dx = Symbol('du_dx')\n", "dv_dx = Symbol('dv_dx')\n", "dw_dx = Symbol('dw_dx')\n", "dp_dx = Symbol('dP_dx')\n",
+				"ds_dx = Symbol('ds_dx')\n", "\n", "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
+				"L = simplify(lambda_waves * Sinv * dQ_dx_def)\n", "L"
+			]
+		},
+		{"cell_type" : "markdown", "metadata" : {}, "source" : ["### x1 lower inflow (with relaxation term)"]}, {
+			"cell_type" : "code",
+			"execution_count" : 7,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "Hi0PadhkikAAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - "
+							"dP_{dx}\\right)\\\\\\frac{R c \\eta_{2} \\rho \\left(t - t_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{3} "
+							"\\left(v - v_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{4} \\left(w - "
+							"w_{t}\\right)}{L_{x}}\\\\\\frac{c^{2} \\eta_{5} \\rho \\left(1 - M^{2}\\right) \\left(u - "
+							"u_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{6} \\left(s - s_{t}\\right)}{L_{x}}\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡(c - u)⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n", "⎢                           ⎥\n", "⎢     R⋅c⋅η₂⋅ρ⋅(t - tₜ)     ⎥\n",
+						"⎢     ─────────────────     ⎥\n", "⎢             Lₓ            ⎥\n", "⎢                           ⎥\n",
+						"⎢       c⋅η₃⋅(v - vₜ)       ⎥\n", "⎢       ─────────────       ⎥\n", "⎢             Lₓ            ⎥\n",
+						"⎢                           ⎥\n", "⎢       c⋅η₄⋅(w - wₜ)       ⎥\n", "⎢       ─────────────       ⎥\n",
+						"⎢             Lₓ            ⎥\n", "⎢                           ⎥\n", "⎢  2      ⎛     2⎞          ⎥\n",
+						"⎢ c ⋅η₅⋅ρ⋅⎝1 - M ⎠⋅(u - uₜ) ⎥\n", "⎢ ───────────────────────── ⎥\n", "⎢             Lₓ            ⎥\n",
+						"⎢                           ⎥\n", "⎢       c⋅η₆⋅(s - sₜ)       ⎥\n", "⎢       ─────────────       ⎥\n",
+						"⎣             Lₓ            ⎦"
+					]
+				},
+				"execution_count" : 7,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"eta_2 = Symbol('eta_2')\n",
+				"eta_3 = Symbol('eta_3')\n",
+				"eta_4 = Symbol('eta_4')\n",
+				"eta_5 = Symbol('eta_5')\n",
+				"eta_6 = Symbol('eta_6')\n",
+				"Lx = Symbol('L_x')\n",
+				"M = Symbol('M')\n",
+				"u_t = Symbol('u_t')\n",
+				"v = Symbol('v')\n",
+				"v_t = Symbol('v_t')\n",
+				"w = Symbol('w')\n",
+				"w_t = Symbol('w_t')\n",
+				"s = Symbol('s')\n",
+				"s_t = Symbol('s_t')\n",
+				"t = Symbol('t') # gas temperature\n",
+				"t_t = Symbol('t_t')\n",
+				"R = Symbol('R') # gas constant (kB/mu)\n",
+				"L_inflow_x1_lower = Matrix([L[0],\n",
+				"                            eta_2*(c/Lx)*(rho*R)*(t-t_t),\n",
+				"                            eta_3*(c/Lx)*(v-v_t),\n",
+				"                            eta_4*(c/Lx)*(w-w_t),\n",
+				"                            eta_5*(rho*c**2 * (1 - M**2) / Lx)*(u - u_t),\n",
+				"                            eta_6*(c/Lx)*(s-s_t)])\n",
+				"L_inflow_x1_lower"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 8,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}\\frac{L_{x} u \\left(c + u\\right) \\left(- c du_{dx} \\rho + "
+							"dP_{dx}\\right) - 2 R c \\eta_{2} \\rho \\left(c + u\\right) \\left(t - t_{t}\\right) - c^{2} "
+							"\\eta_{5} \\rho u \\left(M^{2} - 1\\right) \\left(u - u_{t}\\right)}{2 L_{x} c^{2} u \\left(c + "
+							"u\\right)}\\\\\\frac{L_{x} \\left(c + u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right) - c^{2} "
+							"\\eta_{5} \\rho \\left(M^{2} - 1\\right) \\left(u - u_{t}\\right)}{2 L_{x} c \\rho \\left(c + "
+							"u\\right)}\\\\\\frac{c \\eta_{3} \\left(v - v_{t}\\right)}{L_{x} u}\\\\\\frac{c \\eta_{4} \\left(w - "
+							"w_{t}\\right)}{L_{x} u}\\\\\\frac{L_{x} \\left(c + u\\right) \\left(- c du_{dx} \\rho + "
+							"dP_{dx}\\right) - c^{2} \\eta_{5} \\rho \\left(M^{2} - 1\\right) \\left(u - u_{t}\\right)}{2 L_{x} "
+							"\\left(c + u\\right)}\\\\\\frac{c \\eta_{6} \\left(s - s_{t}\\right)}{L_{x} u}\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                                                                   2        ⎛\n",
+						"⎢Lₓ⋅u⋅(c + u)⋅(-c⋅du_dx⋅ρ + dP_dx) - 2⋅R⋅c⋅η₂⋅ρ⋅(c + u)⋅(t - tₜ) - c ⋅η₅⋅ρ⋅u⋅⎝\n",
+						"⎢─────────────────────────────────────────────────────────────────────────────\n",
+						"⎢                                            2                                \n",
+						"⎢                                      2⋅Lₓ⋅c ⋅u⋅(c + u)                      \n",
+						"⎢                                                                             \n",
+						"⎢                                                   2      ⎛ 2    ⎞           \n",
+						"⎢                 Lₓ⋅(c + u)⋅(c⋅du_dx⋅ρ - dP_dx) - c ⋅η₅⋅ρ⋅⎝M  - 1⎠⋅(u - uₜ)  \n",
+						"⎢                 ──────────────────────────────────────────────────────────  \n",
+						"⎢                                      2⋅Lₓ⋅c⋅ρ⋅(c + u)                       \n",
+						"⎢                                                                             \n",
+						"⎢                                        c⋅η₃⋅(v - vₜ)                        \n",
+						"⎢                                        ─────────────                        \n",
+						"⎢                                             Lₓ⋅u                            \n",
+						"⎢                                                                             \n",
+						"⎢                                        c⋅η₄⋅(w - wₜ)                        \n",
+						"⎢                                        ─────────────                        \n",
+						"⎢                                             Lₓ⋅u                            \n",
+						"⎢                                                                             \n",
+						"⎢                                                    2      ⎛ 2    ⎞          \n",
+						"⎢                 Lₓ⋅(c + u)⋅(-c⋅du_dx⋅ρ + dP_dx) - c ⋅η₅⋅ρ⋅⎝M  - 1⎠⋅(u - uₜ) \n",
+						"⎢                 ─────────────────────────────────────────────────────────── \n",
+						"⎢                                         2⋅Lₓ⋅(c + u)                        \n",
+						"⎢                                                                             \n",
+						"⎢                                        c⋅η₆⋅(s - sₜ)                        \n",
+						"⎢                                        ─────────────                        \n",
+						"⎣                                             Lₓ⋅u                            \n",
+						"\n",
+						" 2    ⎞         ⎤\n",
+						"M  - 1⎠⋅(u - uₜ)⎥\n",
+						"────────────────⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎥\n",
+						"                ⎦"
+					]
+				},
+				"execution_count" : 8,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "dQ_dx_inflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_inflow_x1_lower)\n", "dQ_dx_inflow_x1_lower" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 9,
+			"metadata" : {},
+			"outputs" : [ {
+				"name" : "stdout",
+				"output_type" : "stream",
+				"text" : [
+					"dQ_dx[0] = (1.0/2.0)*(L_x*u*(c + u)*(-c*du_dx*rho + dP_dx) - 2*R*c*eta_2*rho*(c + u)*(t - t_t) - std::pow(c, "
+					"2)*eta_5*rho*u*(std::pow(M, 2) - 1)*(u - u_t))/(L_x*std::pow(c, 2)*u*(c + u));\n",
+					"dQ_dx[1] = (1.0/2.0)*(L_x*(c + u)*(c*du_dx*rho - dP_dx) - std::pow(c, 2)*eta_5*rho*(std::pow(M, 2) - 1)*(u - "
+					"u_t))/(L_x*c*rho*(c + u));\n",
+					"dQ_dx[2] = c*eta_3*(v - v_t)/(L_x*u);\n", "dQ_dx[3] = c*eta_4*(w - w_t)/(L_x*u);\n",
+					"dQ_dx[4] = (1.0/2.0)*(L_x*(c + u)*(-c*du_dx*rho + dP_dx) - std::pow(c, 2)*eta_5*rho*(std::pow(M, 2) - 1)*(u - "
+					"u_t))/(L_x*(c + u));\n",
+					"dQ_dx[5] = c*eta_6*(s - s_t)/(L_x*u);\n"
+				]
+			} ],
+			"source" : [
+				"drho_dx = Symbol('drho_dx')\n", "du_dx = Symbol('du_dx')\n", "dv_dx = Symbol('dv_dx')\n", "dw_dx = Symbol('dw_dx')\n",
+				"dp_dx = Symbol('dP_dx')\n", "ds_dx = Symbol('ds_dx')\n", "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
+				"L = simplify(lambda_waves * Sinv * dQ_dx_def)\n", "\n", "L_inflow_x1_lower = Matrix([L[0],\n",
+				"                            eta_2*(c/Lx)*(rho*R)*(t-t_t),\n", "                            eta_3*(c/Lx)*(v-v_t),\n",
+				"                            eta_4*(c/Lx)*(w-w_t),\n",
+				"                            eta_5*(rho*c**2 * (1 - M**2) / Lx)*(u - u_t),\n",
+				"                            eta_6*(c/Lx)*(s-s_t)])\n", "\n",
+				"dQ_dx_inflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_inflow_x1_lower)\n",
+				"print(cxxcode(dQ_dx_inflow_x1_lower, assign_to='dQ_dx'))"
+			]
+		},
+		{"cell_type" : "markdown", "metadata" : {}, "source" : ["### x1 lower inflow (with relaxation term; zero normal velocity)"]}, {
+			"cell_type" : "code",
+			"execution_count" : 10,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "ogf77Y3ip/60R2wu0/0XjKK+EieIZeEAAAAASUVORK5CYII=",
+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}c \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\\\frac{R c \\eta_{2} \\rho "
+					     "\\left(t - t_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{3} \\left(v - v_{t}\\right)}{L_{x}}\\\\\\frac{c \\eta_{4} "
+					     "\\left(w - w_{t}\\right)}{L_{x}}\\\\- \\frac{c^{2} \\eta_{5} \\rho u_{t} \\left(1 - "
+					     "M^{2}\\right)}{L_{x}}\\\\\\frac{c \\eta_{6} \\left(s - s_{t}\\right)}{L_{x}}\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡c⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n", "⎢                     ⎥\n", "⎢  R⋅c⋅η₂⋅ρ⋅(t - tₜ)  ⎥\n",
+						"⎢  ─────────────────  ⎥\n", "⎢          Lₓ         ⎥\n", "⎢                     ⎥\n",
+						"⎢    c⋅η₃⋅(v - vₜ)    ⎥\n", "⎢    ─────────────    ⎥\n", "⎢          Lₓ         ⎥\n",
+						"⎢                     ⎥\n", "⎢    c⋅η₄⋅(w - wₜ)    ⎥\n", "⎢    ─────────────    ⎥\n",
+						"⎢          Lₓ         ⎥\n", "⎢                     ⎥\n", "⎢  2         ⎛     2⎞ ⎥\n",
+						"⎢-c ⋅η₅⋅ρ⋅uₜ⋅⎝1 - M ⎠ ⎥\n", "⎢─────────────────────⎥\n", "⎢          Lₓ         ⎥\n",
+						"⎢                     ⎥\n", "⎢    c⋅η₆⋅(s - sₜ)    ⎥\n", "⎢    ─────────────    ⎥\n",
+						"⎣          Lₓ         ⎦"
+					]
+				},
+				"execution_count" : 10,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "L_inflow_x1_lower_zerou = L_inflow_x1_lower.subs(u, 0)\n", "L_inflow_x1_lower_zerou" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 11,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}- \\frac{1}{c} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & "
+							"0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & \\frac{1}{c} & 0\\\\0 & 0 & 0 "
+							"& 0 & 0 & 0\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡-1                ⎤\n", "⎢───  0  0  0  0  0⎥\n", "⎢ c                ⎥\n", "⎢                  ⎥\n",
+						"⎢ 0   0  0  0  0  0⎥\n", "⎢                  ⎥\n", "⎢ 0   0  0  0  0  0⎥\n", "⎢                  ⎥\n",
+						"⎢ 0   0  0  0  0  0⎥\n", "⎢                  ⎥\n", "⎢              1   ⎥\n", "⎢ 0   0  0  0  ─  0⎥\n",
+						"⎢              c   ⎥\n", "⎢                  ⎥\n", "⎣ 0   0  0  0  0  0⎦"
+					]
+				},
+				"execution_count" : 11,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"def pinv(x): # Moore-Penrose pseudoinverse of a diagonal matrix\n", "    if x != 0:\n", "        return x**-1\n",
+				"    else:\n", "        return 0\n", "ll = [pinv(l.subs(u,0)) for l in lambda_list]\n",
+				"linv_zerou = diag(ll[1], ll[0], ll[0], ll[0], ll[2], ll[0])\n", "linv_zerou"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 12,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "VER4uhOyWNQ8edh7X0PmN8rnTwkOkvGrxRmmb7CuIl9OFIuBG6UIVPyE2L1qn27qKh4fhVIenpBMpnafSefj2RmF4FzBlesQRWIiAL98WAnfm1fjKA"
+					    "l6QEe+"
+					    "u8UBc7bWI9G6jFfbQEViDgBulNeidb128pMH3FAuxMV7hKxBFup86AosQ8OXbItjOu5I8ovIVAYwU37zqfIoi85wWvXh53ii6dEsRcE9pKXIXUi8aH"
+					    "jaya1/ExEsqv1l0Ici4mHsh4EZpL2TPgK8M0mOJwX7S0Bcxq/"
+					    "tJqseXHvI9qTNAw0U4FgK+fDsW0vesnWiQ+Cxw+OJhPD8ozN9rwvCku3SZiDxO8IpzlWdz/BMd3Lkb/"
+					    "Fa78pwcgYCAe0o+EHoIyJDgIfEYAN+34V9cMCx87M2+tnmIaUrqfcCMpR536uyZpjnf8Yaf04Uj4J7ShQ+AAfH56B8PUBImkqEJXpFC3mp/"
+					    "FjN4K5wo5fGIMGDBu1J4UJ5tmuNVTX2rnSpOF46AG6ULHwA18WVI/"
+					    "lRLtzTl9z5ub3llqLL8WwvlMVR4Wwed8xyUGSuSnByBhIAbpQSFR3ZCoPyON95U6zNQO3XJ2Z4yAm6UTlk7Z9A3eUT++"
+					    "skZ6PGYIvhG9zHR9rYcAUdgEgE3SpMQeQFHwBE4JgK+"
+					    "fDsm2mfYlpZnPFwJ8QwSzyLxhYH8WSbynByBZgTcU2qGyguWCMj48MgA31DiIUvusHF37QfFec7JyRFYhIAbpUWweSUZHv5MkddJ+"
+					    "LhbIMV5YJJzHrx0cgQWIeBGaRFsXkkIcFet89WAiMobhXz0jYcpnRyB2Qi4UZoNmVeICNwoTK+"
+					    "dZKjYfhL5To7AbATcKM2GzCs0ekGPHClHYAkCbpSWoOZ1zOCwhzREvnwbQsbTRxFwozQKj2euQODDFXW96gUj4EbpgpW/"
+					    "QvTaXpKxMy+K55acHIHZCLhRmg2ZV4i3/gGitkSzNNvwdsAcgVkIuFGaBZcXzhDgzwIeZ+cWNU/J/"
+					    "0zAEPFwFgJulGbB5YUzBL5TnM+QlMQfDPyceVNlvp87AqMIuFEahcczhxCQ0eEjbe8U8vnbQIqzdHuq4/n7FP91BOYj4C/kzsfMa/"
+					    "yBAF4RL+A+UcjGNuGnOq896a0sJ0dgGgE3StMYeYkBBGR8bpVV+zeTgRqe7AhMI+"
+					    "DLt2mMvIQj4AgcEQE3SkcE25tyBByBaQTcKE1j5CUcAUfgiAi4UToi2N6UI+AITCPgRmkaIy/hCDgCR0TAjdIRwfamHAFHYBqB/"
+					    "JGAX3SLt6zxvdL8f7tKVPzcEXAEFiMgm/KLKj8eYoBR4sXJoWdN/KXKIeQ83RFwBJYiYP+AU63//2MYId96fmtzAAAAAElFTkSuQmCC",
+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}\\frac{L_{x} \\left(- c du_{dx} \\rho + dP_{dx}\\right) + c \\eta_{5} "
+					     "\\rho u_{t} \\left(M^{2} - 1\\right)}{2 L_{x} c^{2}}\\\\\\frac{L_{x} \\left(c du_{dx} \\rho - dP_{dx}\\right) + "
+					     "c \\eta_{5} \\rho u_{t} \\left(M^{2} - 1\\right)}{2 L_{x} c \\rho}\\\\0\\\\0\\\\\\frac{L_{x} \\left(- c du_{dx} "
+					     "\\rho + dP_{dx}\\right) + c \\eta_{5} \\rho u_{t} \\left(M^{2} - 1\\right)}{2 L_{x}}\\\\0\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                                    ⎛ 2    ⎞⎤\n", "⎢Lₓ⋅(-c⋅du_dx⋅ρ + dP_dx) + c⋅η₅⋅ρ⋅uₜ⋅⎝M  - 1⎠⎥\n",
+						"⎢────────────────────────────────────────────⎥\n", "⎢                        2                   ⎥\n",
+						"⎢                  2⋅Lₓ⋅c                    ⎥\n", "⎢                                            ⎥\n",
+						"⎢                                   ⎛ 2    ⎞ ⎥\n", "⎢Lₓ⋅(c⋅du_dx⋅ρ - dP_dx) + c⋅η₅⋅ρ⋅uₜ⋅⎝M  - 1⎠ ⎥\n",
+						"⎢─────────────────────────────────────────── ⎥\n", "⎢                  2⋅Lₓ⋅c⋅ρ                  ⎥\n",
+						"⎢                                            ⎥\n", "⎢                     0                      ⎥\n",
+						"⎢                                            ⎥\n", "⎢                     0                      ⎥\n",
+						"⎢                                            ⎥\n", "⎢                                    ⎛ 2    ⎞⎥\n",
+						"⎢Lₓ⋅(-c⋅du_dx⋅ρ + dP_dx) + c⋅η₅⋅ρ⋅uₜ⋅⎝M  - 1⎠⎥\n", "⎢────────────────────────────────────────────⎥\n",
+						"⎢                    2⋅Lₓ                    ⎥\n", "⎢                                            ⎥\n",
+						"⎣                     0                      ⎦"
+					]
+				},
+				"execution_count" : 12,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "dQ_dx_inflow_x1_lower_zerou = simplify(S * linv_zerou * L_inflow_x1_lower_zerou)\n", "dQ_dx_inflow_x1_lower_zerou" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 13,
+			"metadata" : {},
+			"outputs" : [ {
+				"name" : "stdout",
+				"output_type" : "stream",
+				"text" : [
+					"dQ_dx[0] = (1.0/2.0)*(L_x*(-c*du_dx*rho + dP_dx) + c*eta_5*rho*u_t*(std::pow(M, 2) - 1))/(L_x*std::pow(c, 2));\n",
+					"dQ_dx[1] = (1.0/2.0)*(L_x*(c*du_dx*rho - dP_dx) + c*eta_5*rho*u_t*(std::pow(M, 2) - 1))/(L_x*c*rho);\n",
+					"dQ_dx[2] = 0;\n", "dQ_dx[3] = 0;\n",
+					"dQ_dx[4] = (1.0/2.0)*(L_x*(-c*du_dx*rho + dP_dx) + c*eta_5*rho*u_t*(std::pow(M, 2) - 1))/L_x;\n", "dQ_dx[5] = 0;\n"
+				]
+			} ],
+			"source" : ["print(cxxcode(dQ_dx_inflow_x1_lower_zerou, assign_to='dQ_dx'))"]
+		}
+	],
+	"metadata" : {
+		"kernelspec" : {"display_name" : "Python 3", "language" : "python", "name" : "python3"},
+		"language_info" : {
+			"codemirror_mode" : {"name" : "ipython", "version" : 3},
+			"file_extension" : ".py",
+			"mimetype" : "text/x-python",
+			"name" : "python",
+			"nbconvert_exporter" : "python",
+			"pygments_lexer" : "ipython3",
+			"version" : "3.11.5"
+		},
+		"orig_nbformat" : 4
+	},
+	"nbformat" : 4,
+	"nbformat_minor" : 2
 }
diff --git a/src/problems/NSCBC/dQ_dx_outflow.ipynb b/src/problems/NSCBC/dQ_dx_outflow.ipynb
index 361d0573f..aa4463cd3 100644
--- a/src/problems/NSCBC/dQ_dx_outflow.ipynb
+++ b/src/problems/NSCBC/dQ_dx_outflow.ipynb
@@ -1,970 +1,2957 @@
 {
-	"cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Navier-Stokes Characteristic Boundary Conditions (NSCBC)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "from sympy import *\n",
-    "init_printing(use_unicode=True)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Conserved variables\n",
-    "We start with the Euler equations in conservation-law form, with the vector of conserved variables $U$ and primitive variables $q$. Then we compute the $P$ matrix ($P_{ij} = \\partial U_i / \\partial q_j$) and the $Q_x$, $Q_y$, and $Q_z$ matrices ($Q_{k,ij} = \\partial F_{k,i} / \\partial q_j$, where $F_k$ is the flux vector in the $k$ direction) (Sutherland and Kennedy, 2002)."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle \\frac{P - c^{2} \\rho^{2} \\frac{\\partial}{\\partial P} e{\\left(\\rho,P \\right)}}{\\rho^{2}}$"
-      ],
-      "text/plain": [
-       "     2  2 ∂          \n",
-       "P - c ⋅ρ ⋅──(e(ρ, P))\n",
-       "          ∂P         \n",
-       "─────────────────────\n",
-       "           2         \n",
-       "          ρ          "
-      ]
-     },
-     "execution_count": 2,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# define symbols\n",
-    "rho = Symbol('rho', positive=True) # density\n",
-    "p = Symbol('P', positive=True) # pressure\n",
-    "u = Symbol('u') # normal velocity\n",
-    "v = Symbol('v') # transverse velocity\n",
-    "w = Symbol('w') # transverse velocity\n",
-    "s = Symbol('s') # passive scalar\n",
-    "c = Symbol('c') # sound speed\n",
-    "e = Function('e')(rho, p) # specific internal energy\n",
-    "\n",
-    "E = rho*e + rho*(u**2 + v**2 + w**2)/2 # total energy density\n",
-    "\n",
-    "# (this is needed to rewrite the thermodynamic derivatives in terms of the sound speed)\n",
-    "de_drho = solveset(Eq(c**2, (p - rho**2 * Derivative(e, rho)) / (rho**2 * Derivative(e,p))), Derivative(e, rho)).args[0]\n",
-    "de_drho"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0\\\\u & \\rho & 0 & 0 & 0 & 0\\\\v & 0 & \\rho & 0 & 0 & 0\\\\w & 0 & 0 & \\rho & 0 & 0\\\\\\rho \\frac{\\partial}{\\partial \\rho} e{\\left(\\rho,P \\right)} + \\frac{u^{2}}{2} + \\frac{v^{2}}{2} + \\frac{w^{2}}{2} + e{\\left(\\rho,P \\right)} & \\rho u & \\rho v & \\rho w & \\rho \\frac{\\partial}{\\partial P} e{\\left(\\rho,P \\right)} & 0\\\\s & 0 & 0 & 0 & 0 & \\rho\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                  1                      0    0    0         0        0⎤\n",
-       "⎢                                                                       ⎥\n",
-       "⎢                  u                      ρ    0    0         0        0⎥\n",
-       "⎢                                                                       ⎥\n",
-       "⎢                  v                      0    ρ    0         0        0⎥\n",
-       "⎢                                                                       ⎥\n",
-       "⎢                  w                      0    0    ρ         0        0⎥\n",
-       "⎢                                                                       ⎥\n",
-       "⎢                 2    2    2                                           ⎥\n",
-       "⎢  ∂             u    v    w                              ∂             ⎥\n",
-       "⎢ρ⋅──(e(ρ, P)) + ── + ── + ── + e(ρ, P)  ρ⋅u  ρ⋅v  ρ⋅w  ρ⋅──(e(ρ, P))  0⎥\n",
-       "⎢  ∂ρ            2    2    2                              ∂P            ⎥\n",
-       "⎢                                                                       ⎥\n",
-       "⎣                  s                      0    0    0         0        ρ⎦"
-      ]
-     },
-     "execution_count": 3,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the P matrix from Sutherland and Kennedy (2003)\n",
-    "U = Matrix([rho, rho*u, rho*v, rho*w, E, rho*s])\n",
-    "q = Matrix([rho, u, v, w, p, s])\n",
-    "P = simplify(Matrix(BlockMatrix(Derivative(U, q, evaluate=True).T)).T)\n",
-    "P"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# compute the Q_n matrix\n",
-    "F_n = Matrix([rho*u,  # flux vector in normal direction\n",
-    "            rho*u**2 + p,\n",
-    "            rho*v*u,\n",
-    "            rho*w*u,\n",
-    "            (E + p)*u,\n",
-    "            rho*s*u])\n",
-    "Q_n = simplify(Matrix(BlockMatrix(Derivative(F_n, q, evaluate=True).T)).T)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# compute the Q_t1 matrix\n",
-    "F_t1 = Matrix([rho*v,\n",
-    "              rho*u*v,\n",
-    "              rho*v**2 + p,\n",
-    "              rho*w*v,\n",
-    "              (E + p)*v,\n",
-    "              rho*s*v])\n",
-    "Q_t1 = simplify(Matrix(BlockMatrix(Derivative(F_t1, q, evaluate=True).T)).T)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# compute the Q_t2 matrix\n",
-    "F_t2 = Matrix([rho*w,\n",
-    "               rho*u*w,\n",
-    "               rho*v*w,\n",
-    "               rho*w**2 + p,\n",
-    "               (E + p)*w,\n",
-    "               rho*s*w])\n",
-    "Q_t2 = simplify(Matrix(BlockMatrix(Derivative(F_t2, q, evaluate=True).T)).T)\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Primitive variables\n",
-    "Now we compute the coefficient matrices $A_x$, $A_y$, and $A_z$ for the primitive equations ($A = P^{-1} Q$):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 7,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}u & \\rho & 0 & 0 & 0 & 0\\\\0 & u & 0 & 0 & \\frac{1}{\\rho} & 0\\\\0 & 0 & u & 0 & 0 & 0\\\\0 & 0 & 0 & u & 0 & 0\\\\0 & c^{2} \\rho & 0 & 0 & u & 0\\\\0 & 0 & 0 & 0 & 0 & u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡u   ρ    0  0  0  0⎤\n",
-       "⎢                   ⎥\n",
-       "⎢               1   ⎥\n",
-       "⎢0   u    0  0  ─  0⎥\n",
-       "⎢               ρ   ⎥\n",
-       "⎢                   ⎥\n",
-       "⎢0   0    u  0  0  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎢0   0    0  u  0  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎢    2              ⎥\n",
-       "⎢0  c ⋅ρ  0  0  u  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎣0   0    0  0  0  u⎦"
-      ]
-     },
-     "execution_count": 7,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the A_n matrix from Sutherland and Kennedy (2003)\n",
-    "# (this is the primitive variable flux matrix in the normal direction)\n",
-    "A_n = simplify(Inverse(P) * Q_n).subs(Derivative(e,rho), de_drho)\n",
-    "A_n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 8,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}v & 0 & \\rho & 0 & 0 & 0\\\\0 & v & 0 & 0 & 0 & 0\\\\0 & 0 & v & 0 & \\frac{1}{\\rho} & 0\\\\0 & 0 & 0 & v & 0 & 0\\\\0 & 0 & c^{2} \\rho & 0 & v & 0\\\\0 & 0 & 0 & 0 & 0 & v\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡v  0   ρ    0  0  0⎤\n",
-       "⎢                   ⎥\n",
-       "⎢0  v   0    0  0  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎢               1   ⎥\n",
-       "⎢0  0   v    0  ─  0⎥\n",
-       "⎢               ρ   ⎥\n",
-       "⎢                   ⎥\n",
-       "⎢0  0   0    v  0  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎢       2           ⎥\n",
-       "⎢0  0  c ⋅ρ  0  v  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎣0  0   0    0  0  v⎦"
-      ]
-     },
-     "execution_count": 8,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the A_t1 matrix\n",
-    "A_t1 = simplify(Inverse(P) * Q_t1).subs(Derivative(e,rho), de_drho)\n",
-    "A_t1"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}w & 0 & 0 & \\rho & 0 & 0\\\\0 & w & 0 & 0 & 0 & 0\\\\0 & 0 & w & 0 & 0 & 0\\\\0 & 0 & 0 & w & \\frac{1}{\\rho} & 0\\\\0 & 0 & 0 & c^{2} \\rho & w & 0\\\\0 & 0 & 0 & 0 & 0 & w\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡w  0  0   ρ    0  0⎤\n",
-       "⎢                   ⎥\n",
-       "⎢0  w  0   0    0  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎢0  0  w   0    0  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎢               1   ⎥\n",
-       "⎢0  0  0   w    ─  0⎥\n",
-       "⎢               ρ   ⎥\n",
-       "⎢                   ⎥\n",
-       "⎢          2        ⎥\n",
-       "⎢0  0  0  c ⋅ρ  w  0⎥\n",
-       "⎢                   ⎥\n",
-       "⎣0  0  0   0    0  w⎦"
-      ]
-     },
-     "execution_count": 9,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the A_t2 matrix\n",
-    "A_t2 = simplify(Inverse(P) * Q_t2).subs(Derivative(e,rho), de_drho)\n",
-    "A_t2"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Characteristic variables\n",
-    "We compute the eigenvalues $\\lambda$ and eigenvectors $S$ of the $A_x$ matrix.\n",
-    "\n",
-    "Then we compute the vector $\\mathcal{L} = \\lambda S^{-1} \\frac{dq}{dx}$ and the transverse vectors $\\mathcal{T}_y = S^{-1} A_y \\frac{dq}{dy}$ and $\\mathcal{T}_z = S^{-1} A_z \\frac{dq}{dz}$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}0 & - c \\rho & 0 & 0 & 1 & 0\\\\- c^{2} & 0 & 0 & 0 & 1 & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & 0\\\\0 & c \\rho & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 0   -c⋅ρ  0  0  1  0⎤\n",
-       "⎢                     ⎥\n",
-       "⎢  2                  ⎥\n",
-       "⎢-c    0    0  0  1  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎢ 0    0    1  0  0  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎢ 0    0    0  1  0  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎢ 0   c⋅ρ   0  0  1  0⎥\n",
-       "⎢                     ⎥\n",
-       "⎣ 0    0    0  0  0  1⎦"
-      ]
-     },
-     "execution_count": 10,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the eigenvector matrix S\n",
-    "c = Symbol('c')\n",
-    "rho = Symbol('rho')\n",
-    "lambda_list = [eval for (eval, n, es) in A_n.left_eigenvects()]\n",
-    "Sinv_list = [[ev for ev in es] for (v, n, es) in A_n.left_eigenvects()]\n",
-    "Sinv = Matrix([Sinv_list[1][0],  Sinv_list[0][2], Sinv_list[0][0], Sinv_list[0][1], Sinv_list[2][0], Sinv_list[0][3]])\n",
-    "Sinv"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}- c + u & 0 & 0 & 0 & 0 & 0\\\\0 & u & 0 & 0 & 0 & 0\\\\0 & 0 & u & 0 & 0 & 0\\\\0 & 0 & 0 & u & 0 & 0\\\\0 & 0 & 0 & 0 & c + u & 0\\\\0 & 0 & 0 & 0 & 0 & u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡-c + u  0  0  0    0    0⎤\n",
-       "⎢                         ⎥\n",
-       "⎢  0     u  0  0    0    0⎥\n",
-       "⎢                         ⎥\n",
-       "⎢  0     0  u  0    0    0⎥\n",
-       "⎢                         ⎥\n",
-       "⎢  0     0  0  u    0    0⎥\n",
-       "⎢                         ⎥\n",
-       "⎢  0     0  0  0  c + u  0⎥\n",
-       "⎢                         ⎥\n",
-       "⎣  0     0  0  0    0    u⎦"
-      ]
-     },
-     "execution_count": 11,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute eigenvalue matrix \\lambda\n",
-    "lambda_waves = diag(lambda_list[1], lambda_list[0], lambda_list[0], lambda_list[0], lambda_list[2], lambda_list[0])\n",
-    "lambda_waves"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 12,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2 c^{2}} & - \\frac{1}{c^{2}} & 0 & 0 & \\frac{1}{2 c^{2}} & 0\\\\- \\frac{1}{2 c \\rho} & 0 & 0 & 0 & \\frac{1}{2 c \\rho} & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & 0\\\\\\frac{1}{2} & 0 & 0 & 0 & \\frac{1}{2} & 0\\\\0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 1     -1          1      ⎤\n",
-       "⎢────   ───  0  0  ────   0⎥\n",
-       "⎢   2     2           2    ⎥\n",
-       "⎢2⋅c     c         2⋅c     ⎥\n",
-       "⎢                          ⎥\n",
-       "⎢ -1                 1     ⎥\n",
-       "⎢─────   0   0  0  ─────  0⎥\n",
-       "⎢2⋅c⋅ρ             2⋅c⋅ρ   ⎥\n",
-       "⎢                          ⎥\n",
-       "⎢  0     0   1  0    0    0⎥\n",
-       "⎢                          ⎥\n",
-       "⎢  0     0   0  1    0    0⎥\n",
-       "⎢                          ⎥\n",
-       "⎢ 1/2    0   0  0   1/2   0⎥\n",
-       "⎢                          ⎥\n",
-       "⎣  0     0   0  0    0    1⎦"
-      ]
-     },
-     "execution_count": 12,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute S\n",
-    "S = Sinv.inv()\n",
-    "S"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 13,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\u \\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\\\left(c + u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡(c - u)⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n",
-       "⎢                           ⎥\n",
-       "⎢   ⎛   2                ⎞  ⎥\n",
-       "⎢ u⋅⎝- c ⋅drho_dx + dP_dx⎠  ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢          dv_dx⋅u          ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢          dw_dx⋅u          ⎥\n",
-       "⎢                           ⎥\n",
-       "⎢(c + u)⋅(c⋅du_dx⋅ρ + dP_dx)⎥\n",
-       "⎢                           ⎥\n",
-       "⎣          ds_dx⋅u          ⎦"
-      ]
-     },
-     "execution_count": 13,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the \\mathcal{L} vector\n",
-    "\n",
-    "#drho_dx = Symbol(r'\\frac{d\\rho}{dx}')\n",
-    "#du_dx = Symbol(r'\\frac{du}{dx}')\n",
-    "#dv_dx = Symbol(r'\\frac{dv}{dx}')\n",
-    "#dw_dx = Symbol(r'\\frac{dw}{dx}')\n",
-    "#dp_dx = Symbol(r'\\frac{dP}{dx}')\n",
-    "#ds_dx = Symbol(r'\\frac{ds}{dx}')\n",
-    "\n",
-    "drho_dx = Symbol('drho_dx')\n",
-    "du_dx = Symbol('du_dx')\n",
-    "dv_dx = Symbol('dv_dx')\n",
-    "dw_dx = Symbol('dw_dx')\n",
-    "dp_dx = Symbol('dP_dx')\n",
-    "ds_dx = Symbol('ds_dx')\n",
-    "\n",
-    "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
-    "L = simplify(lambda_waves * Sinv * dQ_dx_def)\n",
-    "L"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 14,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}c^{2} dv_{dy} \\rho - c du_{dy} \\rho v + dP_{dy} v\\\\v \\left(- c^{2} drho_{dy} + dP_{dy}\\right)\\\\\\frac{dP_{dy}}{\\rho} + dv_{dy} v\\\\dw_{dy} v\\\\c^{2} dv_{dy} \\rho + c du_{dy} \\rho v + dP_{dy} v\\\\ds_{dy} v\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 2                                ⎤\n",
-       "⎢c ⋅dv_dy⋅ρ - c⋅du_dy⋅ρ⋅v + dP_dy⋅v⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢       ⎛   2                ⎞     ⎥\n",
-       "⎢     v⋅⎝- c ⋅drho_dy + dP_dy⎠     ⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢         dP_dy                    ⎥\n",
-       "⎢         ───── + dv_dy⋅v          ⎥\n",
-       "⎢           ρ                      ⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢             dw_dy⋅v              ⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢ 2                                ⎥\n",
-       "⎢c ⋅dv_dy⋅ρ + c⋅du_dy⋅ρ⋅v + dP_dy⋅v⎥\n",
-       "⎢                                  ⎥\n",
-       "⎣             ds_dy⋅v              ⎦"
-      ]
-     },
-     "execution_count": 14,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the y-transverse terms\n",
-    "\n",
-    "#drho_dy = Symbol(r'\\frac{d\\rho}{dy}')\n",
-    "#du_dy = Symbol(r'\\frac{du}{dy}')\n",
-    "#dv_dy = Symbol(r'\\frac{dv}{dy}')\n",
-    "#dw_dy = Symbol(r'\\frac{dw}{dy}')\n",
-    "#dp_dy = Symbol(r'\\frac{dP}{dy}')\n",
-    "#ds_dy = Symbol(r'\\frac{ds}{dy}')\n",
-    "\n",
-    "drho_dy = Symbol('drho_dy')\n",
-    "du_dy = Symbol('du_dy')\n",
-    "dv_dy = Symbol('dv_dy')\n",
-    "dw_dy = Symbol('dw_dy')\n",
-    "dp_dy = Symbol('dP_dy')\n",
-    "ds_dy = Symbol('ds_dy')\n",
-    "\n",
-    "dq_dy_def = Matrix([drho_dy, du_dy, dv_dy, dw_dy, dp_dy, ds_dy])\n",
-    "T_1 = simplify(Sinv * A_t1 * dq_dy_def)\n",
-    "T_1"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 15,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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HVLZmifzbWgslslXCuqCTBlIj0aAA2g2gGDbZwE8cggn+Mp639hhAR3Nmycc8vLrxlZ6By+BK64hJzMD8QQ7iSY8NYtLhpjA4suBgJAJfsxrSJy7UhzwdqUzK4J2ddEGV+jGN6w1OXfcsQ13zhIcwk9ssmo7/micqexauiSxYVQflr1F4a2FB+2ItUodN0vNNSiWhBBpPJ7jDsTgFgOyV+FjuXRJW2guhJEUKSkl5WbRs2iDiFx6ryqcj2frMC52/XVAWyog6vpFvfwlkQPNYljDwMKKDMZWZ6uTIwnQiJxQCltwruU456DytD7jz1Ify83KwcFLliaLP0ygotCeKsFdOiDjfoRpX1ROMwRaZD7pGgfIYvbTQTBJoLSxYqytiGkrdwGGzSgRs1GilO1baaashFC9b+MQaad4o4t8NwmqhCgnFBwVXquNRGHVSWnusW+D2FKR0PavCEip8cFAobrQ+ikcx4VIrhHIYbIQf5FMXfJRLtyCqawboo8WTZm2KZR1hqHKPwpS21O8GRVT6NbGgv6Y3jkE5LhVxd6mCL1QuTzCKA+pC8uy22Dgog5KgEroOloZOUWzclQMpHIWBhYbSOPpJ2FOq6iPldWVW51o5oerVk6sVFuLzQqKDK/12s7RpS2QF1NDo7AcZvcuuUO61suQuzrSTxVXWfsD1tRzTTBvshP1KDsXyTg7L5ejur7BJEs/mFuRkofUJ1sACzNjAaFjWS3PGlDf38yo1CGsD7+V3jynPiLcX5QhUI6A+yvpat5BenfEMCSUbT+pYs3x2a9MZ4P1C7oETJ0dgqwgkgzRdc9qkuDenRNQ4mIY8NWGe7uQIbA4B9U3WkXitYnDhe0tC35wSAXw1DlMZ5u08NXByBLaGAOtIi9aNLlGRW1tY7TCWAgk7NbsAP3EENoKA+uasR8yXFhtLhDkXQrO67uQIOAKOQA0CPCkLyg4lwvyLR5/3ck6OgCPgCNQgwFJA2AR3s9OZGpT2mEamMJuT2I/BzWH2/2n2WGeX+bIIuBK5LP7NS5cS4R2hxf+naS6QM7x6BJjOOF0ZAlIkmJrQlnd4Pknox90j4Epk901YrADvB41+qKiYywMdgQUIuBJZANoOsmCJuBWyg4a6BhF9TWTnraipCwuo7L7lUT3f/kB5sLj6Vi6Q0vA2KO86lP7jw/sZvETX7F2iifKQje+VhMeD8vnoEvLP/q2o8jhtAAG3RDbQCEtF0ADE4uANWpQAjm3Stp0/tUQeYhwfWLZ4nQbi0T5v3raksfJ4bwnFZ8RrCFzz3QynHSLgSmSHjYbIUgpYF7yRjPJIX9LinN8nMDhJx50f5QGFtZKn0+7IC4lp/i5iyUmhvHwTI4qPL4UFUnqUHU+UmsnwxNmP50LApzPnQrp9OVgUfOox377PIE2nJoP/8YkijQ5e8aecN3K1P6MaLC8qGJRfaiUhBr8aHZWDRE7bRMAtkW22S41UfLCmNxg1EBmgTA3SO71ZJKw9EH+kdJQvVTpK0qOwK1FpsBYmSenGykPB9X5RERnad2kn+XuC7SGAEvlZFMv87UnpEvUQiMoChdApi5gAxXJQfE+5xLihx74sto4RyqfEbywPcZTXTatiYsJ6vCQr060qBRV5uLcNBD4yMZjO/DdemG9x7m8fgXwKEAYuYmtwMvgZxJYGC6U3WGOaYGmQB1IYyonFT6wDzll0fS/XkdIw8FmPwerglXV4kzdVEIRZ2URDhOVWz2vl49V3p30h8G8T985O3N8PAhp0DF4GbBi8SK4wlAaD+weuRfln9VAgaXrO+QJ7N9B1jtLgaQ+fvWPaQxnwTJWDLsPbm+Gbonau9HkayoNfIMUzlbmXS2UgLLemQno/7AeBm/vG6n6aZlzSOODfKRVPXuz/NCgXFkIJ4xcZueWB1cFADgNX8b31EV3b+oft4UCB/KjwZ/KPSOEoBPZ8FC0J46c0yMUeFspD5kc5wub8N0jJnbaCgNq2+8Yq0xmnHSKgRmQQlr5+VQoLNVSeoBxGqsuaSpofSyG3MIwXCuaVeBYVCIkGykv5B15+2DcCd/sW36VvhYAG/AvxwnXTG52zxhJ+RhXjdRmUA8rlpcLCN0DlMy0ir9MNIuBK5AYbvVRlKQEsG1wgXQdLQxdYIuGpDxEKZwrDlKnFz6hg6bRzBHw6s/MGbCw+Uw12wLK4yroFU5XXcqv8jEp8na4AAVciV9CIraog5YHVUVwDsTKUxtZVWCTtLcxaGvdvCwGfztxWe3ttHYHmCLgSaQ6pM3QEbgsBVyK31d5eW0egOQK+JtIIUq0VFN92VTiPQ1mgxGcNgScgPA5l0xeUv8r/FOpHR2AnCKBE2BfAYhmr8U7LEWC35xspjXyXKDtHeSzKXgpblAyl6BrFw+7RT0OAHxyB/SDAAnzoz0xn6OC2HXo/VdiepGNvux69vRrFZys4+Ds5AntDAMs6vCaBJeK0AAFZEUxJHuQG33ZN2AL4F8m1naLJ87daLc59R2AXCGCJOM1EICqQmrddD0qLpYHC6fZfECbHq/RMdY7eJVHYH+Tg7+QIbB4Bt0SWNRFrGSiA9D2Tg6576yGRNVYIi6lfKj4GhXWoL3RNeIlYJ0Hx7IpUH7bK85YuitN/4bmr1lsurCuRZdhVv+0q9mE9RAMsvKxWWVx48a0y7WaSqY4oUf+F52Za5DyCuBKZibMGChYCLrVCwqCPcQf5qYWBJTL4urwVrzx8nwGCN4u0YdGKAEjxWD+Es4bCguw3WTkKqiflRS6mVU23rke+CNJN3+ql8pR7RMDXRGa2Why4nZLQdfFtV9gq7mg9pFSc0qFAPpWPYsBiuZefKimy8bUxHgWzkMtUqpOByAWEssK1JhSq/8KzNaob5udKZFnj2NuuDH4Uhb3terDBLf+Nws2a4OtfYwOWpzQhrdKhlOwThzp9IoUzMCkPBVJae7Gkl/axcNwKuXQrnLF8n84sAFuDmEEyOlCUBouidh0EBfMYRWEQ8iGgz+W6x786ZzrzVj6/XECZsHD5Iea5iKfyUaDIhdXEFAtMUIJv5QIpDXVD3tIvPEm7+i81tyBDAONKD26JbKNhsWT49SRrHhDTlvtwpoPCsWqYxvxZ5/iEXVqBoOx4DM22fRwKE4UCpQp2C7/U3IIMT8hc4dEtkQ00qgYgFkdndeQixQFKcNNF0Lyc2mvJg3XBPpf8vR8sku5fM0qHpcFHoyHWSszaCgE6oIhSq8Wmavl6kKU/iCeKCqVa9Ue+NWTohPGTgABK5GcRC/MdmhEE1Cl/Gok+e5TkKX6J3QRRPGstDNacgqWjeNZjckIRHG2CSxIxkE/6haf4o2BQRqnVQhFTv9SkPkfvKJFxgPy3ngPAnBj8keVHifw3Xphvce4XEFDnHx20hSwXDRpQEgeFM3XiEW/tuk1aj9/qojf4xQeFwBoJG+UCKSxMuWJZxKeWFIptyS81kbtXdihs4LCSDAOl3VTwv622KBEnR6AagagsUAidsoiZUSwHxZcGOFOZbpoT0xPWS6u8WCd/j/HBi+U96GLwHaWYj+kVSgsLCmV2UHjKf7EM8HIaRsAXVoex8ZhxBPJ1izBIyaLBy5OlMJAjC87z9KWw1+nA1znKisVb9shgxaAUUDSpctBleCU9PHa385QPCUSl8kphPRmesvpxDIG7sUiPcwRyBDQ4udsziBmAgRTGFIPBbftbSr/wRCEEUnqmMvdyKQ/CcuuGtZfJd5TEj8fEWDBBUXGt85yIXyJDzsevMwR8OpMB4pdVCHDXf6fBylMS+4UnlghvHxMWNs7JD6QwnuJgTRCOEmJPycdy8LCw0i81mSKlC7womtwKUVCwflBirygrBGQHwikrKa9WhoyTX+YIuBLJETnDtToyd0Tm+XTkz+TYRNZbC1DYZkmyogjSwW2ylsJCnPKUrIOx9GCES6dBKKrwRz6YRjnwUS7dIrGusXD4uRZydqTrWTJ0Gf1kFIG70ViPXAuBv4rxe3Vqnoy8leP63MQA6w2ycwswVl5UAJ18ug6WhvJgiYRFXPIrHIXBtAel8Qs5FAoWR5dX104rIuCWyIrgllirc2PusxfCLA/utOy5wJ2t46us4rSgJPMFw7BUmIawuPrIudxrOf8jn0DYCrkSOX9LYFKnZvVLRDinAjl/lZeVGBXdqLJTGsOSpzfpPpRlhXqu2Qj4dGY2ZMszqMO/UG7Mb3uKATPutoNb3kng5AhsGQG3RM7bOszXD1ImYdoiH4VC2C8Jd3IE9oiAK5HzthpPF76T8mBdBEXC27q8SHa2tRCV5+QINEUAJcLCHvNKFq6c1kWA9Q8e5/r0ZV2cnfv6CLBWFdajWBPBpGbDDzsIndZFgMeUowuF6xbv3B2BZggwDUdvHFAiTmdAIK5/tPg26hmk9SIcgXoEmq2JaJDw5IFdmBDWDTT2b5WnFDdyFD5MG1kTcXIErgqBZkpEqPDehD2zP+gcU4dNQv6z6qvqMl4ZR6CPQMvpzJdSHMyTjNiKzPsMrAM4OQKOwJUi0NISwQpJN1FVQxYVjf9+sRqx+Qkd4/mYnZJjCm/F85if6a3deFlwt0f9LA3woIPxxLZ/C9fl9qiZElFF8y3HKJX0+5aDtVde3iPx3y8OInR6hGN8OoZzOEzhrXhevvyjfL7Zy1ffjtbLFMb3Vf4l/2O5zSqSltOZDmNVmCkMH6qp3ompPKlG7nhd2wn1lOM/LGenW8F4CbBrtMsU3oq3qf77AZlZV8QquUh/GZDpKLi5EhEwn6gU1kPm7sREE9/C7xfpFLhL0K1gvATbNdplCu+pGydjCeLDT5ulZtMZahgVCHO4YJrFa8LTD8sMgQGgvhFrCJ024Y5xGxxruUzhbePEPguR8w3xChyyVPL0F7merUSkELAy+CIX9EEufBsjKgzML5SImWmsixx9ri6mhQ/KBV4oD/K8lQukNNwZLvr7xShK50kmOgVv3VJv7hLUtUZBKmlbUrnFdrBSFG8WYQ3GPIb/UXl661q65gvqq+/1UTmDuCpuc/3AME59yTmJd5o+ng8qGfFjPBBPHxtSMgWWw0FrYfl8uMh+TASJhR4qFd79kI/SYFChMdkTQoPjd6Q0KJKOdA0wdE6mO2EAyrcP9KaWyIPCKYu1FZ7cpB38QdeUZWSDmndTUh4Wf7IvORi0fDgo1Ec+9aCuPz+Z+QwGKpfOOtYOhyjbSRiLBwqcr7bTvquR+E/huql+UAJiBt5dduWxG631/RCncNqXPka//42uW75ntQqW1UpEFaKyvIGaVorP1AXrQeGTg0lpGPh07vwOjjLp/kuidAD8NzkIBZW/HMgADuWSQOmb/34RvkbiT4MyoNKNczx+Q6mwULqK4rLyM3+qHVphDO6Td0DVHSXA40puCpPp07oo/SiuSku7N+0HafktzlWHKrwLZdGHoc8ihk9XT//X4TuyRxa8JYjpZ2GuPM3HlMlTpUQkABYHGrIbuDBQ+KTiIF1CdtdJrQqiATRVTumj4U5RkVBlAgYNlw9cplVjUwvqMOf3i0re0Tud5Y0KHhCyHJFkoTzrKGn8PReK526TE4p08M4feU61w8kYR6GQvdfeubDxek1cw1Q5ltOkH0QMW7ZLLd45fGG9Y6y98wzJ9RLM1xhTQaTniWBjpzRgi5fHAp+0IIHIIGRgdGadwj6QRj53KuJTpUMH+KC4/K7HH9LGCF654hlLH+JUDuUhw7dZYuuIuRyWr6QkrE7dl8kznlOXNe0wF2OrX4ex6kx7UOcavFbDVXI07wfi2bpdqvAuNCy4F/tOIW0eNBvzNbA0oZ7byZCvwulMuG6QD6UdCx/hQyMcFF/qsMGkVlzoTJE/Yb20isc66TWIwpD5Qe7U3y9iGXRTLZ0b8cFgtPuY9WNpT/ZjfUbbYSTNGMbUL3+0Tic9iF8P0xh2CVwX9wNkXosW4g2u9Feo14+fgo6PsZwWfRnmzbG8Oxa5H6IK2AA2v5dA8XZH7oWPXOSDLlSK9OK1xd8vUj/WPzqSnHQC3ODUo4tbPHQAAAZASURBVEvc6ERlGv7m9zgrPm2HuRjnyqJT1OL7pdwLCos+i8n8kgHLhUEADvlg4G4PNkwBw7nS52nm4IpllNepFPa6UI6yrk65bGN9GmGsrSZvzI0xp+wSbqWwaizv4FpBdJijAaMKMh98rMh/UFo6Px0JgQMpjDsendAG6aZ+vyj5GDzIey8XKIaxOGy/bowxZ/FG22Ehxr0BIB60x0s5UyyfRr5UkPY++beWEcM5uCJLUGQIofwMQtok7UuETQ5KpWlGEZe5fZrysWKpR65YCc6pCeYJ0+ZYPk+YD56qsgwYXvWnQrb2QKPyqT+UQy2hiPb0+0W7Y/A06U2sJE9oePRmg6y27ienU5k17TALYwmFtUCb0Lb/kUOp8LqC/RLzvc6NmBalNxPwKQ4E8UMZvZIP/5xm4QoPOawfFhTpb8hZ8xvOvNw1rqvxjvKj+MDmoGuUHk9ieI9miFphHvirrOZYVikRSqfwoVrWhosHHSDthJa1FBbilKe0EDaWHuWGS++wmJc0FuGHKAc+nblb5NQ1Dfxo8ToPZqmu4TXW0Io+D0mW0XaIspfwKYUdatMr3UVxVfmz+sF5WqPrSyVsj8IG6jAoamPMu3IG5DiSt8swcXI3Eb+7aAH0QULjAuk63BF1wR0zLDASoXAUBndflMbQ7xcH77TwWEg9+RbyOHs2YdSTG8wkhOGzBVxPxaRXv1OZtcjfGPMWIhV5VFsixdzbDUSrYraxCPjIudxruerfLyrvC6VH0dhmJ52eTuJbNP9P53wWDpvF9dTab7hdTsb8VGym8j/76quvuKMw2FhAS6cAU3mvOl5YcJdlzvpznXOXcmqAgOPaAMQNsFA78loE61TPrm460xBfnhjxtMgVSENQxcpxbYvnxbld63TmZGCj8tjz1ONkDNZg4Liugeplebolcln8vXRHYPcIuBLZfRN6BRyByyLgSuSy+HvpjsDuEXAlsvsm9Ao4ApdFwJXIZfH30h2B3SPgT2d234T9CujpB/t+3sl9IvetrkvbxfuZ/MoROAEBVyIngLfFrFIavBjoPwLbYuNcqUw+nbnChpUiYbct5PtcnnDw44oIuBJZEdwLsubt4/xrZRcUx4u+ZgRciVxn62KJuBVynW27uVr5msjmmmSeQJq6sIDKJw14eZKP9aA8WFx9K3dESm8fHyKODyzxtnPT94PEjzegN/XjMSrrtA4Cbomsg+tZuGqwYnHwBjaKAMeHk1AS0JElonjeSn5PupiWzzxaevK0In6ShCx8RiHnzweHUXxGKDCu+SSj0w4RcCWyw0ZDZA1S7vYoAZRH+gkHzo++Tq80DNRX8tPPOvKdFZTQICk9n0n8SQ7rZpJiOvsGC2szfM8lJRQfyiyQ0qPskCmtw1OkH3eBgE9ndtFMRSG5w/MHvu5/MTEVg/S7Yg4FKj0DGOXDB5cZ5FPEd03n/PRrtZ8kTQnq8ZdBwC2Ry+DeolQ+SdibskgpYJ1gcXR3eitIcdzpmWKgZFAMfOUtn2oo+Ij4In+vnKMUSYB4Mj05yCcf8qRKjrKX/HhM2Zy2ioBbIlttmRG5NEAZnLhcWYRvnSq+OOgVzvSF6Q+DmXMsjN4X+3UNX9YtTv3pl33k+oN4GRHWk03lMU1Kp1iW1v2dIOCWyE4aakDMfB0hDFzSanB2PwLTOesa9qsP4opTGYWjQFgjafFzKiyiXL5S2GvkUVqnnSLgSmSHDadBx92dgcegDKQwpg/c1Us/ArMpzFNiHWP6byIvC2d6c/LPqSIzrAuUUiCVgwz3cqnMhOXWVEjvh/0g8Hw/orqkGQJ8Bbz2R2BfKO0rDWT2btjAZm0ifzmP6RB8jRjkRStBeVFY8GRadESEy231h1NH8nrAcgRciSzH7qI5NUCxRtIBb/IchSktVsHouoPSoFxw6RSE6dGSn34pW7B2ciVF8JF8BDrtF4G7/YrukrdEIColFFMgXQdLQxdYInN/TvXExI83gYBbIjfRzNWVxEpgGsLi6iPncq/lqn/6pbRON4aAK5Eba/Cx6kp5YHUU10Asn9LYFIX9H+keEEvi/o0hkCoR7jZ59b9TmM9hc1T82hG4IQSkA9ge0D1Vy6uOEmEhze4ueXy6yJbH+bUj4AjcBgKjO5v/HwKeLUfK0MYJAAAAAElFTkSuQmCC",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}c^{2} dw_{dz} \\rho - c du_{dz} \\rho w + dP_{dz} w\\\\w \\left(- c^{2} drho_{dz} + dP_{dz}\\right)\\\\dv_{dz} w\\\\\\frac{dP_{dz}}{\\rho} + dw_{dz} w\\\\c^{2} dw_{dz} \\rho + c du_{dz} \\rho w + dP_{dz} w\\\\ds_{dz} w\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 2                                ⎤\n",
-       "⎢c ⋅dw_dz⋅ρ - c⋅du_dz⋅ρ⋅w + dP_dz⋅w⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢       ⎛   2                ⎞     ⎥\n",
-       "⎢     w⋅⎝- c ⋅drho_dz + dP_dz⎠     ⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢             dv_dz⋅w              ⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢         dP_dz                    ⎥\n",
-       "⎢         ───── + dw_dz⋅w          ⎥\n",
-       "⎢           ρ                      ⎥\n",
-       "⎢                                  ⎥\n",
-       "⎢ 2                                ⎥\n",
-       "⎢c ⋅dw_dz⋅ρ + c⋅du_dz⋅ρ⋅w + dP_dz⋅w⎥\n",
-       "⎢                                  ⎥\n",
-       "⎣             ds_dz⋅w              ⎦"
-      ]
-     },
-     "execution_count": 15,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute the z-transverse terms\n",
-    "\n",
-    "#drho_dz = Symbol(r'\\frac{d\\rho}{dz}')\n",
-    "#du_dz = Symbol(r'\\frac{du}{dz}')\n",
-    "#dv_dz = Symbol(r'\\frac{dv}{dz}')\n",
-    "#dw_dz = Symbol(r'\\frac{dw}{dz}')\n",
-    "#dp_dz = Symbol(r'\\frac{dP}{dz}')\n",
-    "#ds_dz = Symbol(r'\\frac{ds}{dz}')\n",
-    "\n",
-    "drho_dz = Symbol('drho_dz')\n",
-    "du_dz = Symbol('du_dz')\n",
-    "dv_dz = Symbol('dv_dz')\n",
-    "dw_dz = Symbol('dw_dz')\n",
-    "dp_dz = Symbol('dP_dz')\n",
-    "ds_dz = Symbol('ds_dz')\n",
-    "\n",
-    "dq_dz_def = Matrix([drho_dz, du_dz, dv_dz, dw_dz, dp_dz, ds_dz])\n",
-    "T_2 = simplify(Sinv * A_t2 * dq_dz_def)\n",
-    "T_2"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### x1 Upper Outflow\n",
-    "We replace the $\\mathcal{L}_0$ component with a relaxation term for a specified pressure at the outflow boundary to obtain $\\mathcal{\\tilde L}$, and then compute ${d\\tilde Q}/{dx} = S \\lambda^{-1} \\mathcal{\\tilde L}$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 16,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}K \\left(P - P_{t}\\right) - \\left(1 - \\beta\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right)\\\\u \\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\\\left(c + u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                     ⎛ 2            2                                        \n",
-       "⎢K⋅(P - Pₜ) - (1 - β)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz⋅ρ⋅w + d\n",
-       "⎢                                                                             \n",
-       "⎢                                     ⎛   2                ⎞                  \n",
-       "⎢                                   u⋅⎝- c ⋅drho_dx + dP_dx⎠                  \n",
-       "⎢                                                                             \n",
-       "⎢                                           dv_dx⋅u                           \n",
-       "⎢                                                                             \n",
-       "⎢                                           dw_dx⋅u                           \n",
-       "⎢                                                                             \n",
-       "⎢                                 (c + u)⋅(c⋅du_dx⋅ρ + dP_dx)                 \n",
-       "⎢                                                                             \n",
-       "⎣                                           ds_dx⋅u                           \n",
-       "\n",
-       "                ⎞⎤\n",
-       "P_dy⋅v + dP_dz⋅w⎠⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎦"
-      ]
-     },
-     "execution_count": 16,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "K = Symbol('K')\n",
-    "p = Symbol('P')\n",
-    "p_t = Symbol('P_t')\n",
-    "beta = Symbol('beta')\n",
-    "v = Symbol('v')\n",
-    "dp_dy = Symbol('dP_dy')\n",
-    "du_dy = Symbol('du_dy')\n",
-    "gamma = Symbol('gamma')\n",
-    "dv_dy = Symbol('dv_dy')\n",
-    "\n",
-    "T_phi = simplify(T_1[0] + T_2[0])\n",
-    "L_outflow_x1_upper = Matrix([K*(p - p_t) - (1 - beta)*T_phi, L[1], L[2], L[3], L[4], L[5]])\n",
-    "L_outflow_x1_upper"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\frac{- K \\left(P - P_{t}\\right) - \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c - u\\right) \\left(2 c^{2} drho_{dx} + c du_{dx} \\rho - dP_{dx}\\right)}{2 c^{2} \\left(c - u\\right)}\\\\\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c - u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)}{2 c \\rho \\left(c - u\\right)}\\\\dv_{dx}\\\\dw_{dx}\\\\\\frac{- K \\left(P - P_{t}\\right) - \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c - u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)}{2 \\left(c - u\\right)}\\\\ds_{dx}\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                      ⎛ 2            2                                       \n",
-       "⎢-K⋅(P - Pₜ) - (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz⋅ρ⋅w + \n",
-       "⎢─────────────────────────────────────────────────────────────────────────────\n",
-       "⎢                                                                   2         \n",
-       "⎢                                                                2⋅c ⋅(c - u) \n",
-       "⎢                                                                             \n",
-       "⎢                             ⎛ 2            2                                \n",
-       "⎢        K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz\n",
-       "⎢        ─────────────────────────────────────────────────────────────────────\n",
-       "⎢                                                               2⋅c⋅ρ⋅(c - u) \n",
-       "⎢                                                                             \n",
-       "⎢                                                                   dv_dx     \n",
-       "⎢                                                                             \n",
-       "⎢                                                                   dw_dx     \n",
-       "⎢                                                                             \n",
-       "⎢                             ⎛ 2            2                                \n",
-       "⎢       -K⋅(P - Pₜ) - (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz\n",
-       "⎢       ──────────────────────────────────────────────────────────────────────\n",
-       "⎢                                                                 2⋅(c - u)   \n",
-       "⎢                                                                             \n",
-       "⎣                                                                   ds_dx     \n",
-       "\n",
-       "                 ⎞           ⎛   2                            ⎞⎤\n",
-       "dP_dy⋅v + dP_dz⋅w⎠ + (c - u)⋅⎝2⋅c ⋅drho_dx + c⋅du_dx⋅ρ - dP_dx⎠⎥\n",
-       "───────────────────────────────────────────────────────────────⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                        ⎞                                      ⎥\n",
-       "⋅ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c - u)⋅(c⋅du_dx⋅ρ + dP_dx)        ⎥\n",
-       "───────────────────────────────────────────────────────        ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                        ⎞                                      ⎥\n",
-       "⋅ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c - u)⋅(c⋅du_dx⋅ρ + dP_dx)        ⎥\n",
-       "───────────────────────────────────────────────────────        ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎥\n",
-       "                                                               ⎦"
-      ]
-     },
-     "execution_count": 17,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "dQ_dx_outflow_x1_upper = simplify(S * Inverse(lambda_waves) * L_outflow_x1_upper)\n",
-    "dQ_dx_outflow_x1_upper"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 18,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x0 = std::pow(c, 2);\n",
-      "x1 = c - u;\n",
-      "x2 = c*rho;\n",
-      "x3 = du_dx*x2;\n",
-      "x4 = rho*x0;\n",
-      "x5 = K*(P - P_t) + (beta - 1)*(dP_dy*v + dP_dz*w - du_dy*v*x2 - du_dz*w*x2 + dv_dy*x4 + dw_dz*x4);\n",
-      "x6 = (1.0/2.0)/x1;\n",
-      "x7 = dP_dx + x3;\n",
-      "dQ_dx[0] = x6*(x1*(-dP_dx + 2*drho_dx*x0 + x3) - x5)/x0;\n",
-      "dQ_dx[1] = x6*(x1*x7 + x5)/(c*rho);\n",
-      "dQ_dx[2] = dv_dx;\n",
-      "dQ_dx[3] = dw_dx;\n",
-      "dQ_dx[4] = x6*(x1*x7 - x5);\n",
-      "dQ_dx[5] = ds_dx;\n"
-     ]
-    }
-   ],
-   "source": [
-    "drho_dx = Symbol('drho_dx')\n",
-    "du_dx = Symbol('du_dx')\n",
-    "dv_dx = Symbol('dv_dx')\n",
-    "dw_dx = Symbol('dw_dx')\n",
-    "dp_dx = Symbol('dP_dx')\n",
-    "ds_dx = Symbol('ds_dx')\n",
-    "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
-    "L = simplify(lambda_waves * Sinv * dQ_dx_def)\n",
-    "\n",
-    "T_phi = simplify(T_1[0] + T_2[0])\n",
-    "L_outflow_x1_upper = Matrix([K*(p - p_t) - (1 - beta)*T_phi, L[1], L[2], L[3], L[4], L[5]])\n",
-    "dQ_dx_outflow_x1_upper = simplify(S * Inverse(lambda_waves) * L_outflow_x1_upper)\n",
-    "\n",
-    "common_exprs_upper, final_expr_upper = cse(dQ_dx_outflow_x1_upper, list=False)\n",
-    "for var, val in common_exprs_upper:\n",
-    "    print(cxxcode(val, assign_to=var))\n",
-    "print(cxxcode(final_expr_upper, assign_to='dQ_dx'))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### x1 Lower Outflow\n",
-    "We replace the $\\mathcal{L}_4$ component with a relaxation term for a specified pressure at the outflow boundary to obtain $\\mathcal{\\tilde L}$, and then compute ${d\\tilde Q}/{dx} = S \\lambda^{-1} \\mathcal{\\tilde L}$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 19,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\u \\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\K \\left(P - P_{t}\\right) - \\left(1 - \\beta\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                                 (c - u)⋅(c⋅du_dx⋅ρ - dP_dx)                 \n",
-       "⎢                                                                             \n",
-       "⎢                                     ⎛   2                ⎞                  \n",
-       "⎢                                   u⋅⎝- c ⋅drho_dx + dP_dx⎠                  \n",
-       "⎢                                                                             \n",
-       "⎢                                           dv_dx⋅u                           \n",
-       "⎢                                                                             \n",
-       "⎢                                           dw_dx⋅u                           \n",
-       "⎢                                                                             \n",
-       "⎢                     ⎛ 2            2                                        \n",
-       "⎢K⋅(P - Pₜ) - (1 - β)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅ρ⋅w + d\n",
-       "⎢                                                                             \n",
-       "⎣                                           ds_dx⋅u                           \n",
-       "\n",
-       "                 ⎤\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                 ⎥\n",
-       "                ⎞⎥\n",
-       "P_dy⋅v + dP_dz⋅w⎠⎥\n",
-       "                 ⎥\n",
-       "                 ⎦"
-      ]
-     },
-     "execution_count": 19,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "K = Symbol('K')\n",
-    "p = Symbol('P')\n",
-    "p_t = Symbol('P_t')\n",
-    "beta = Symbol('beta')\n",
-    "v = Symbol('v')\n",
-    "dp_dy = Symbol('dP_dy')\n",
-    "du_dy = Symbol('du_dy')\n",
-    "gamma = Symbol('gamma')\n",
-    "dv_dy = Symbol('dv_dy')\n",
-    "\n",
-    "T_phi = simplify(T_1[4] + T_2[4])\n",
-    "L_outflow_x1_lower = Matrix([L[0], L[1], L[2], L[3], K*(p - p_t) - (1 - beta)*T_phi, L[5]])\n",
-    "L_outflow_x1_lower"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 20,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c + u\\right) \\left(2 c^{2} drho_{dx} - c du_{dx} \\rho - dP_{dx}\\right)}{2 c^{2} \\left(c + u\\right)}\\\\\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c + u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right)}{2 c \\rho \\left(c + u\\right)}\\\\dv_{dx}\\\\dw_{dx}\\\\\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c + u\\right) \\left(- c du_{dx} \\rho + dP_{dx}\\right)}{2 \\left(c + u\\right)}\\\\ds_{dx}\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡                     ⎛ 2            2                                        \n",
-       "⎢K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅ρ⋅w + d\n",
-       "⎢─────────────────────────────────────────────────────────────────────────────\n",
-       "⎢                                                                   2         \n",
-       "⎢                                                                2⋅c ⋅(c + u) \n",
-       "⎢                                                                             \n",
-       "⎢                            ⎛ 2            2                                 \n",
-       "⎢       K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅\n",
-       "⎢       ──────────────────────────────────────────────────────────────────────\n",
-       "⎢                                                              2⋅c⋅ρ⋅(c + u)  \n",
-       "⎢                                                                             \n",
-       "⎢                                                                   dv_dx     \n",
-       "⎢                                                                             \n",
-       "⎢                                                                   dw_dx     \n",
-       "⎢                                                                             \n",
-       "⎢                            ⎛ 2            2                                 \n",
-       "⎢       K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅\n",
-       "⎢       ──────────────────────────────────────────────────────────────────────\n",
-       "⎢                                                                 2⋅(c + u)   \n",
-       "⎢                                                                             \n",
-       "⎣                                                                   ds_dx     \n",
-       "\n",
-       "                ⎞           ⎛   2                            ⎞⎤\n",
-       "P_dy⋅v + dP_dz⋅w⎠ + (c + u)⋅⎝2⋅c ⋅drho_dx - c⋅du_dx⋅ρ - dP_dx⎠⎥\n",
-       "──────────────────────────────────────────────────────────────⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                       ⎞                                      ⎥\n",
-       "ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c + u)⋅(c⋅du_dx⋅ρ - dP_dx)        ⎥\n",
-       "──────────────────────────────────────────────────────        ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                       ⎞                                      ⎥\n",
-       "ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c + u)⋅(-c⋅du_dx⋅ρ + dP_dx)       ⎥\n",
-       "───────────────────────────────────────────────────────       ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎥\n",
-       "                                                              ⎦"
-      ]
-     },
-     "execution_count": 20,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "dQ_dx_outflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_outflow_x1_lower)\n",
-    "dQ_dx_outflow_x1_lower"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 21,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x0 = std::pow(c, 2);\n",
-      "x1 = c + u;\n",
-      "x2 = c*rho;\n",
-      "x3 = du_dx*x2;\n",
-      "x4 = rho*x0;\n",
-      "x5 = K*(P - P_t) + (beta - 1)*(dP_dy*v + dP_dz*w + du_dy*v*x2 + du_dz*w*x2 + dv_dy*x4 + dw_dz*x4);\n",
-      "x6 = (1.0/2.0)/x1;\n",
-      "x7 = -dP_dx + x3;\n",
-      "dQ_dx[0] = x6*(x1*(-dP_dx + 2*drho_dx*x0 - x3) + x5)/x0;\n",
-      "dQ_dx[1] = x6*(x1*x7 + x5)/(c*rho);\n",
-      "dQ_dx[2] = dv_dx;\n",
-      "dQ_dx[3] = dw_dx;\n",
-      "dQ_dx[4] = x6*(-x1*x7 + x5);\n",
-      "dQ_dx[5] = ds_dx;\n"
-     ]
-    }
-   ],
-   "source": [
-    "drho_dx = Symbol('drho_dx')\n",
-    "du_dx = Symbol('du_dx')\n",
-    "dv_dx = Symbol('dv_dx')\n",
-    "dw_dx = Symbol('dw_dx')\n",
-    "dp_dx = Symbol('dP_dx')\n",
-    "ds_dx = Symbol('ds_dx')\n",
-    "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
-    "L = simplify(lambda_waves * Sinv * dQ_dx_def)\n",
-    "\n",
-    "T_phi = simplify(T_1[4] + T_2[4])\n",
-    "L_outflow_x1_lower = Matrix([L[0], L[1], L[2], L[3], K*(p - p_t) - (1 - beta)*T_phi, L[5]])\n",
-    "dQ_dx_outflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_outflow_x1_lower)\n",
-    "\n",
-    "common_exprs_lower, final_expr_lower = cse(dQ_dx_outflow_x1_lower, list=False)\n",
-    "for var, val in common_exprs_lower:\n",
-    "    print(cxxcode(val, assign_to=var))\n",
-    "print(cxxcode(final_expr_lower, assign_to='dQ_dx'))"
-   ]
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.11.5"
-  },
-  "orig_nbformat": 4
- },
- "nbformat": 4,
- "nbformat_minor": 2
+	"cells" : [
+		{"cell_type" : "markdown", "metadata" : {}, "source" : ["## Navier-Stokes Characteristic Boundary Conditions (NSCBC)"]},
+		{
+			"cell_type" : "code",
+			"execution_count" : 1,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [ "from sympy import *\n", "init_printing(use_unicode=True)" ]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"### Conserved variables\n",
+				"We start with the Euler equations in conservation-law form, with the vector of conserved variables $U$ and primitive "
+				"variables $q$. Then we compute the $P$ matrix ($P_{ij} = \\partial U_i / \\partial q_j$) and the $Q_x$, $Q_y$, and $Q_z$ "
+				"matrices ($Q_{k,ij} = \\partial F_{k,i} / \\partial q_j$, where $F_k$ is the flux vector in the $k$ direction) (Sutherland "
+				"and Kennedy, 2002)."
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 2,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "ue301tEWqgAjXUxrg0F/eL/"
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+					    "VraflQxkgAAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle \\frac{P - c^{2} \\rho^{2} \\frac{\\partial}{\\partial P} e{\\left(\\rho,P "
+							"\\right)}}{\\rho^{2}}$"],
+					"text/plain" : [
+						"     2  2 ∂          \n", "P - c ⋅ρ ⋅──(e(ρ, P))\n", "          ∂P         \n", "─────────────────────\n",
+						"           2         \n", "          ρ          "
+					]
+				},
+				"execution_count" : 2,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# define symbols\n", "rho = Symbol('rho', positive=True) # density\n", "p = Symbol('P', positive=True) # pressure\n",
+				"u = Symbol('u') # normal velocity\n", "v = Symbol('v') # transverse velocity\n", "w = Symbol('w') # transverse velocity\n",
+				"s = Symbol('s') # passive scalar\n", "c = Symbol('c') # sound speed\n",
+				"e = Function('e')(rho, p) # specific internal energy\n", "\n",
+				"E = rho*e + rho*(u**2 + v**2 + w**2)/2 # total energy density\n", "\n",
+				"# (this is needed to rewrite the thermodynamic derivatives in terms of the sound speed)\n",
+				"de_drho = solveset(Eq(c**2, (p - rho**2 * Derivative(e, rho)) / (rho**2 * Derivative(e,p))), Derivative(e, rho)).args[0]\n",
+				"de_drho"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 3,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0\\\\u & \\rho & 0 & 0 & 0 & 0\\\\v & 0 & "
+							"\\rho & 0 & 0 & 0\\\\w & 0 & 0 & \\rho & 0 & 0\\\\\\rho \\frac{\\partial}{\\partial \\rho} "
+							"e{\\left(\\rho,P \\right)} + \\frac{u^{2}}{2} + \\frac{v^{2}}{2} + \\frac{w^{2}}{2} + "
+							"e{\\left(\\rho,P \\right)} & \\rho u & \\rho v & \\rho w & \\rho \\frac{\\partial}{\\partial P} "
+							"e{\\left(\\rho,P \\right)} & 0\\\\s & 0 & 0 & 0 & 0 & \\rho\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                  1                      0    0    0         0        0⎤\n",
+						"⎢                                                                       ⎥\n",
+						"⎢                  u                      ρ    0    0         0        0⎥\n",
+						"⎢                                                                       ⎥\n",
+						"⎢                  v                      0    ρ    0         0        0⎥\n",
+						"⎢                                                                       ⎥\n",
+						"⎢                  w                      0    0    ρ         0        0⎥\n",
+						"⎢                                                                       ⎥\n",
+						"⎢                 2    2    2                                           ⎥\n",
+						"⎢  ∂             u    v    w                              ∂             ⎥\n",
+						"⎢ρ⋅──(e(ρ, P)) + ── + ── + ── + e(ρ, P)  ρ⋅u  ρ⋅v  ρ⋅w  ρ⋅──(e(ρ, P))  0⎥\n",
+						"⎢  ∂ρ            2    2    2                              ∂P            ⎥\n",
+						"⎢                                                                       ⎥\n",
+						"⎣                  s                      0    0    0         0        ρ⎦"
+					]
+				},
+				"execution_count" : 3,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the P matrix from Sutherland and Kennedy (2003)\n", "U = Matrix([rho, rho*u, rho*v, rho*w, E, rho*s])\n",
+				"q = Matrix([rho, u, v, w, p, s])\n", "P = simplify(Matrix(BlockMatrix(Derivative(U, q, evaluate=True).T)).T)\n", "P"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 4,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [
+				"# compute the Q_n matrix\n", "F_n = Matrix([rho*u,  # flux vector in normal direction\n", "            rho*u**2 + p,\n",
+				"            rho*v*u,\n", "            rho*w*u,\n", "            (E + p)*u,\n", "            rho*s*u])\n",
+				"Q_n = simplify(Matrix(BlockMatrix(Derivative(F_n, q, evaluate=True).T)).T)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 5,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [
+				"# compute the Q_t1 matrix\n", "F_t1 = Matrix([rho*v,\n", "              rho*u*v,\n", "              rho*v**2 + p,\n",
+				"              rho*w*v,\n", "              (E + p)*v,\n", "              rho*s*v])\n",
+				"Q_t1 = simplify(Matrix(BlockMatrix(Derivative(F_t1, q, evaluate=True).T)).T)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 6,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [
+				"# compute the Q_t2 matrix\n", "F_t2 = Matrix([rho*w,\n", "               rho*u*w,\n", "               rho*v*w,\n",
+				"               rho*w**2 + p,\n", "               (E + p)*w,\n", "               rho*s*w])\n",
+				"Q_t2 = simplify(Matrix(BlockMatrix(Derivative(F_t2, q, evaluate=True).T)).T)\n"
+			]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"### Primitive variables\n",
+				"Now we compute the coefficient matrices $A_x$, $A_y$, and $A_z$ for the primitive equations ($A = P^{-1} Q$):"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 7,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "B6YuG4JXZLzHAAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}u & \\rho & 0 & 0 & 0 & 0\\\\0 & u & 0 & 0 & \\frac{1}{\\rho} & "
+							"0\\\\0 & 0 & u & 0 & 0 & 0\\\\0 & 0 & 0 & u & 0 & 0\\\\0 & c^{2} \\rho & 0 & 0 & u & 0\\\\0 & 0 & 0 & "
+							"0 & 0 & u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡u   ρ    0  0  0  0⎤\n", "⎢                   ⎥\n", "⎢               1   ⎥\n", "⎢0   u    0  0  ─  0⎥\n",
+						"⎢               ρ   ⎥\n", "⎢                   ⎥\n", "⎢0   0    u  0  0  0⎥\n", "⎢                   ⎥\n",
+						"⎢0   0    0  u  0  0⎥\n", "⎢                   ⎥\n", "⎢    2              ⎥\n", "⎢0  c ⋅ρ  0  0  u  0⎥\n",
+						"⎢                   ⎥\n", "⎣0   0    0  0  0  u⎦"
+					]
+				},
+				"execution_count" : 7,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the A_n matrix from Sutherland and Kennedy (2003)\n",
+				"# (this is the primitive variable flux matrix in the normal direction)\n",
+				"A_n = simplify(Inverse(P) * Q_n).subs(Derivative(e,rho), de_drho)\n", "A_n"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 8,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}v & 0 & \\rho & 0 & 0 & 0\\\\0 & v & 0 & 0 & 0 & 0\\\\0 & 0 & v "
+							"& 0 & \\frac{1}{\\rho} & 0\\\\0 & 0 & 0 & v & 0 & 0\\\\0 & 0 & c^{2} \\rho & 0 & v & 0\\\\0 & 0 & 0 & "
+							"0 & 0 & v\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡v  0   ρ    0  0  0⎤\n", "⎢                   ⎥\n", "⎢0  v   0    0  0  0⎥\n", "⎢                   ⎥\n",
+						"⎢               1   ⎥\n", "⎢0  0   v    0  ─  0⎥\n", "⎢               ρ   ⎥\n", "⎢                   ⎥\n",
+						"⎢0  0   0    v  0  0⎥\n", "⎢                   ⎥\n", "⎢       2           ⎥\n", "⎢0  0  c ⋅ρ  0  v  0⎥\n",
+						"⎢                   ⎥\n", "⎣0  0   0    0  0  v⎦"
+					]
+				},
+				"execution_count" : 8,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "# compute the A_t1 matrix\n", "A_t1 = simplify(Inverse(P) * Q_t1).subs(Derivative(e,rho), de_drho)\n", "A_t1" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 9,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "wNHhotWv0LW7AAAAABJRU5ErkJggg==",
+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}w & 0 & 0 & \\rho & 0 & 0\\\\0 & w & 0 & 0 & 0 & 0\\\\0 & 0 & w "
+							"& 0 & 0 & 0\\\\0 & 0 & 0 & w & \\frac{1}{\\rho} & 0\\\\0 & 0 & 0 & c^{2} \\rho & w & 0\\\\0 & 0 & 0 & "
+							"0 & 0 & w\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡w  0  0   ρ    0  0⎤\n", "⎢                   ⎥\n", "⎢0  w  0   0    0  0⎥\n", "⎢                   ⎥\n",
+						"⎢0  0  w   0    0  0⎥\n", "⎢                   ⎥\n", "⎢               1   ⎥\n", "⎢0  0  0   w    ─  0⎥\n",
+						"⎢               ρ   ⎥\n", "⎢                   ⎥\n", "⎢          2        ⎥\n", "⎢0  0  0  c ⋅ρ  w  0⎥\n",
+						"⎢                   ⎥\n", "⎣0  0  0   0    0  w⎦"
+					]
+				},
+				"execution_count" : 9,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "# compute the A_t2 matrix\n", "A_t2 = simplify(Inverse(P) * Q_t2).subs(Derivative(e,rho), de_drho)\n", "A_t2" ]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"### Characteristic variables\n", "We compute the eigenvalues $\\lambda$ and eigenvectors $S$ of the $A_x$ matrix.\n", "\n",
+				"Then we compute the vector $\\mathcal{L} = \\lambda S^{-1} \\frac{dq}{dx}$ and the transverse vectors $\\mathcal{T}_y = "
+				"S^{-1} A_y \\frac{dq}{dy}$ and $\\mathcal{T}_z = S^{-1} A_z \\frac{dq}{dz}$."
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 10,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}0 & - c \\rho & 0 & 0 & 1 & 0\\\\- c^{2} & 0 & 0 & 0 & 1 & "
+							"0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & 0\\\\0 & c \\rho & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & "
+							"0 & 1\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡ 0   -c⋅ρ  0  0  1  0⎤\n", "⎢                     ⎥\n", "⎢  2                  ⎥\n",
+						"⎢-c    0    0  0  1  0⎥\n", "⎢                     ⎥\n", "⎢ 0    0    1  0  0  0⎥\n",
+						"⎢                     ⎥\n", "⎢ 0    0    0  1  0  0⎥\n", "⎢                     ⎥\n",
+						"⎢ 0   c⋅ρ   0  0  1  0⎥\n", "⎢                     ⎥\n", "⎣ 0    0    0  0  0  1⎦"
+					]
+				},
+				"execution_count" : 10,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the eigenvector matrix S\n", "c = Symbol('c')\n", "rho = Symbol('rho')\n",
+				"lambda_list = [eval for (eval, n, es) in A_n.left_eigenvects()]\n",
+				"Sinv_list = [[ev for ev in es] for (v, n, es) in A_n.left_eigenvects()]\n",
+				"Sinv = Matrix([Sinv_list[1][0],  Sinv_list[0][2], Sinv_list[0][0], Sinv_list[0][1], Sinv_list[2][0], Sinv_list[0][3]])\n",
+				"Sinv"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 11,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
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+					    "wsY5ol++"
+					    "QJ7OhXEoAJWGOmPDmlSWJmenUQXe643TWz5NE0EHiybSAo6EAWs6Dl5nq4KNhXEonYFNgg1ZNJbCdXjj9JZPeiPoYGHmVnDw8ffqYGPRHIpJsCGLVl"
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+					    "GOg/Nv6I2doc9VuU8Q06JTEgBsTAAf7gI2i4uI8KOhS+0PLingMGL7vcc4yKxYAY2A8D6QPyo4j/D04MQISuHGg1AAAAAElFTkSuQmCC",
+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}- c + u & 0 & 0 & 0 & 0 & 0\\\\0 & u & 0 & 0 & 0 & 0\\\\0 & 0 & u & 0 & 0 "
+					     "& 0\\\\0 & 0 & 0 & u & 0 & 0\\\\0 & 0 & 0 & 0 & c + u & 0\\\\0 & 0 & 0 & 0 & 0 & u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡-c + u  0  0  0    0    0⎤\n", "⎢                         ⎥\n", "⎢  0     u  0  0    0    0⎥\n",
+						"⎢                         ⎥\n", "⎢  0     0  u  0    0    0⎥\n", "⎢                         ⎥\n",
+						"⎢  0     0  0  u    0    0⎥\n", "⎢                         ⎥\n", "⎢  0     0  0  0  c + u  0⎥\n",
+						"⎢                         ⎥\n", "⎣  0     0  0  0    0    u⎦"
+					]
+				},
+				"execution_count" : 11,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute eigenvalue matrix \\lambda\n",
+				"lambda_waves = diag(lambda_list[1], lambda_list[0], lambda_list[0], lambda_list[0], lambda_list[2], lambda_list[0])\n",
+				"lambda_waves"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 12,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "2+"
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+					    "JzrWMeXwz+rXDIwcFrmBxOorpBnceGVCagnZXnTWxuvtvycka3jy+GfVS4WGNAMc7Xwyiz2ZpzkTW9tvNrycya1jm8CfxoMTWN5E6e8m3X+"
+					    "blbTHLYOPC28UosMb3pr49WWn7O7dXwbBDC2bJg4IYuJz05wHI/"
+					    "PJu0v47A9YfxxdnNzc4kMPtz0cPjhuDyOg4FgYCUM4Prn80zPsD8T60qshNtQMxjogoEIDF2YMZQIBuoyEIGhLp9RWzDQBQPjwcef0bfYV+"
+					    "o58viYaaRgIBjoiAFc1z9DnYtDKjEwcKTy0LoO3c1GPERE5AcDK2MgvWJxUu3/AySyMXnbmOhyAAAAAElFTkSuQmCC",
+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2 c^{2}} & - \\frac{1}{c^{2}} & 0 & 0 & \\frac{1}{2 c^{2}} & "
+					     "0\\\\- \\frac{1}{2 c \\rho} & 0 & 0 & 0 & \\frac{1}{2 c \\rho} & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 "
+					     "& 0\\\\\\frac{1}{2} & 0 & 0 & 0 & \\frac{1}{2} & 0\\\\0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡ 1     -1          1      ⎤\n", "⎢────   ───  0  0  ────   0⎥\n", "⎢   2     2           2    ⎥\n",
+						"⎢2⋅c     c         2⋅c     ⎥\n", "⎢                          ⎥\n", "⎢ -1                 1     ⎥\n",
+						"⎢─────   0   0  0  ─────  0⎥\n", "⎢2⋅c⋅ρ             2⋅c⋅ρ   ⎥\n", "⎢                          ⎥\n",
+						"⎢  0     0   1  0    0    0⎥\n", "⎢                          ⎥\n", "⎢  0     0   0  1    0    0⎥\n",
+						"⎢                          ⎥\n", "⎢ 1/2    0   0  0   1/2   0⎥\n", "⎢                          ⎥\n",
+						"⎣  0     0   0  0    0    1⎦"
+					]
+				},
+				"execution_count" : 12,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "# compute S\n", "S = Sinv.inv()\n", "S" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 13,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - "
+							"dP_{dx}\\right)\\\\u \\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\\\left(c "
+							"+ u\\right) \\left(c du_{dx} \\rho + dP_{dx}\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡(c - u)⋅(c⋅du_dx⋅ρ - dP_dx)⎤\n", "⎢                           ⎥\n", "⎢   ⎛   2                ⎞  ⎥\n",
+						"⎢ u⋅⎝- c ⋅drho_dx + dP_dx⎠  ⎥\n", "⎢                           ⎥\n", "⎢          dv_dx⋅u          ⎥\n",
+						"⎢                           ⎥\n", "⎢          dw_dx⋅u          ⎥\n", "⎢                           ⎥\n",
+						"⎢(c + u)⋅(c⋅du_dx⋅ρ + dP_dx)⎥\n", "⎢                           ⎥\n", "⎣          ds_dx⋅u          ⎦"
+					]
+				},
+				"execution_count" : 13,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the \\mathcal{L} vector\n", "\n", "#drho_dx = Symbol(r'\\frac{d\\rho}{dx}')\n",
+				"#du_dx = Symbol(r'\\frac{du}{dx}')\n", "#dv_dx = Symbol(r'\\frac{dv}{dx}')\n", "#dw_dx = Symbol(r'\\frac{dw}{dx}')\n",
+				"#dp_dx = Symbol(r'\\frac{dP}{dx}')\n", "#ds_dx = Symbol(r'\\frac{ds}{dx}')\n", "\n", "drho_dx = Symbol('drho_dx')\n",
+				"du_dx = Symbol('du_dx')\n", "dv_dx = Symbol('dv_dx')\n", "dw_dx = Symbol('dw_dx')\n", "dp_dx = Symbol('dP_dx')\n",
+				"ds_dx = Symbol('ds_dx')\n", "\n", "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
+				"L = simplify(lambda_waves * Sinv * dQ_dx_def)\n", "L"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 14,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "vEwG1LesF3ZpBqRaKw7Ri7JCbUlL3OyACak92n7rF35xFFAJzyKG5Xzq/zNP6830g8F7V5NVk/g/xSm7pkJv7QOI2all9Oe2z77/"
+					    "nQztOjkAZgagEGFGwFDjkhhecnG4Ugf8HQyG8B0WFfIkAAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}c^{2} dv_{dy} \\rho - c du_{dy} \\rho v + dP_{dy} v\\\\v "
+							"\\left(- c^{2} drho_{dy} + dP_{dy}\\right)\\\\\\frac{dP_{dy}}{\\rho} + dv_{dy} v\\\\dw_{dy} "
+							"v\\\\c^{2} dv_{dy} \\rho + c du_{dy} \\rho v + dP_{dy} v\\\\ds_{dy} v\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡ 2                                ⎤\n", "⎢c ⋅dv_dy⋅ρ - c⋅du_dy⋅ρ⋅v + dP_dy⋅v⎥\n",
+						"⎢                                  ⎥\n", "⎢       ⎛   2                ⎞     ⎥\n",
+						"⎢     v⋅⎝- c ⋅drho_dy + dP_dy⎠     ⎥\n", "⎢                                  ⎥\n",
+						"⎢         dP_dy                    ⎥\n", "⎢         ───── + dv_dy⋅v          ⎥\n",
+						"⎢           ρ                      ⎥\n", "⎢                                  ⎥\n",
+						"⎢             dw_dy⋅v              ⎥\n", "⎢                                  ⎥\n",
+						"⎢ 2                                ⎥\n", "⎢c ⋅dv_dy⋅ρ + c⋅du_dy⋅ρ⋅v + dP_dy⋅v⎥\n",
+						"⎢                                  ⎥\n", "⎣             ds_dy⋅v              ⎦"
+					]
+				},
+				"execution_count" : 14,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the y-transverse terms\n", "\n", "#drho_dy = Symbol(r'\\frac{d\\rho}{dy}')\n",
+				"#du_dy = Symbol(r'\\frac{du}{dy}')\n", "#dv_dy = Symbol(r'\\frac{dv}{dy}')\n", "#dw_dy = Symbol(r'\\frac{dw}{dy}')\n",
+				"#dp_dy = Symbol(r'\\frac{dP}{dy}')\n", "#ds_dy = Symbol(r'\\frac{ds}{dy}')\n", "\n", "drho_dy = Symbol('drho_dy')\n",
+				"du_dy = Symbol('du_dy')\n", "dv_dy = Symbol('dv_dy')\n", "dw_dy = Symbol('dw_dy')\n", "dp_dy = Symbol('dP_dy')\n",
+				"ds_dy = Symbol('ds_dy')\n", "\n", "dq_dy_def = Matrix([drho_dy, du_dy, dv_dy, dw_dy, dp_dy, ds_dy])\n",
+				"T_1 = simplify(Sinv * A_t1 * dq_dy_def)\n", "T_1"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 15,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}c^{2} dw_{dz} \\rho - c du_{dz} \\rho w + dP_{dz} w\\\\w "
+							"\\left(- c^{2} drho_{dz} + dP_{dz}\\right)\\\\dv_{dz} w\\\\\\frac{dP_{dz}}{\\rho} + dw_{dz} "
+							"w\\\\c^{2} dw_{dz} \\rho + c du_{dz} \\rho w + dP_{dz} w\\\\ds_{dz} w\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡ 2                                ⎤\n", "⎢c ⋅dw_dz⋅ρ - c⋅du_dz⋅ρ⋅w + dP_dz⋅w⎥\n",
+						"⎢                                  ⎥\n", "⎢       ⎛   2                ⎞     ⎥\n",
+						"⎢     w⋅⎝- c ⋅drho_dz + dP_dz⎠     ⎥\n", "⎢                                  ⎥\n",
+						"⎢             dv_dz⋅w              ⎥\n", "⎢                                  ⎥\n",
+						"⎢         dP_dz                    ⎥\n", "⎢         ───── + dw_dz⋅w          ⎥\n",
+						"⎢           ρ                      ⎥\n", "⎢                                  ⎥\n",
+						"⎢ 2                                ⎥\n", "⎢c ⋅dw_dz⋅ρ + c⋅du_dz⋅ρ⋅w + dP_dz⋅w⎥\n",
+						"⎢                                  ⎥\n", "⎣             ds_dz⋅w              ⎦"
+					]
+				},
+				"execution_count" : 15,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute the z-transverse terms\n", "\n", "#drho_dz = Symbol(r'\\frac{d\\rho}{dz}')\n",
+				"#du_dz = Symbol(r'\\frac{du}{dz}')\n", "#dv_dz = Symbol(r'\\frac{dv}{dz}')\n", "#dw_dz = Symbol(r'\\frac{dw}{dz}')\n",
+				"#dp_dz = Symbol(r'\\frac{dP}{dz}')\n", "#ds_dz = Symbol(r'\\frac{ds}{dz}')\n", "\n", "drho_dz = Symbol('drho_dz')\n",
+				"du_dz = Symbol('du_dz')\n", "dv_dz = Symbol('dv_dz')\n", "dw_dz = Symbol('dw_dz')\n", "dp_dz = Symbol('dP_dz')\n",
+				"ds_dz = Symbol('ds_dz')\n", "\n", "dq_dz_def = Matrix([drho_dz, du_dz, dv_dz, dw_dz, dp_dz, ds_dz])\n",
+				"T_2 = simplify(Sinv * A_t2 * dq_dz_def)\n", "T_2"
+			]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"### x1 Upper Outflow\n",
+				"We replace the $\\mathcal{L}_0$ component with a relaxation term for a specified pressure at the outflow boundary to obtain "
+				"$\\mathcal{\\tilde L}$, and then compute ${d\\tilde Q}/{dx} = S \\lambda^{-1} \\mathcal{\\tilde L}$."
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 16,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}K \\left(P - P_{t}\\right) - \\left(1 - \\beta\\right) \\left(c^{2} "
+					     "dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right)\\\\u "
+					     "\\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\\\left(c + u\\right) \\left(c du_{dx} "
+					     "\\rho + dP_{dx}\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                     ⎛ 2            2                                        \n",
+						"⎢K⋅(P - Pₜ) - (1 - β)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz⋅ρ⋅w + d\n",
+						"⎢                                                                             \n",
+						"⎢                                     ⎛   2                ⎞                  \n",
+						"⎢                                   u⋅⎝- c ⋅drho_dx + dP_dx⎠                  \n",
+						"⎢                                                                             \n",
+						"⎢                                           dv_dx⋅u                           \n",
+						"⎢                                                                             \n",
+						"⎢                                           dw_dx⋅u                           \n",
+						"⎢                                                                             \n",
+						"⎢                                 (c + u)⋅(c⋅du_dx⋅ρ + dP_dx)                 \n",
+						"⎢                                                                             \n",
+						"⎣                                           ds_dx⋅u                           \n",
+						"\n",
+						"                ⎞⎤\n",
+						"P_dy⋅v + dP_dz⋅w⎠⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎦"
+					]
+				},
+				"execution_count" : 16,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"K = Symbol('K')\n", "p = Symbol('P')\n", "p_t = Symbol('P_t')\n", "beta = Symbol('beta')\n", "v = Symbol('v')\n",
+				"dp_dy = Symbol('dP_dy')\n", "du_dy = Symbol('du_dy')\n", "gamma = Symbol('gamma')\n", "dv_dy = Symbol('dv_dy')\n", "\n",
+				"T_phi = simplify(T_1[0] + T_2[0])\n",
+				"L_outflow_x1_upper = Matrix([K*(p - p_t) - (1 - beta)*T_phi, L[1], L[2], L[3], L[4], L[5]])\n", "L_outflow_x1_upper"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 17,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}\\frac{- K \\left(P - P_{t}\\right) - \\left(\\beta - 1\\right) "
+					     "\\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} "
+					     "w\\right) + \\left(c - u\\right) \\left(2 c^{2} drho_{dx} + c du_{dx} \\rho - dP_{dx}\\right)}{2 c^{2} \\left(c "
+					     "- u\\right)}\\\\\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} "
+					     "dw_{dz} \\rho - c du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c - u\\right) "
+					     "\\left(c du_{dx} \\rho + dP_{dx}\\right)}{2 c \\rho \\left(c - u\\right)}\\\\dv_{dx}\\\\dw_{dx}\\\\\\frac{- K "
+					     "\\left(P - P_{t}\\right) - \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho - c "
+					     "du_{dy} \\rho v - c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c - u\\right) \\left(c du_{dx} "
+					     "\\rho + dP_{dx}\\right)}{2 \\left(c - u\\right)}\\\\ds_{dx}\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                      ⎛ 2            2                                       \n",
+						"⎢-K⋅(P - Pₜ) - (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz⋅ρ⋅w + \n",
+						"⎢─────────────────────────────────────────────────────────────────────────────\n",
+						"⎢                                                                   2         \n",
+						"⎢                                                                2⋅c ⋅(c - u) \n",
+						"⎢                                                                             \n",
+						"⎢                             ⎛ 2            2                                \n",
+						"⎢        K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz\n",
+						"⎢        ─────────────────────────────────────────────────────────────────────\n",
+						"⎢                                                               2⋅c⋅ρ⋅(c - u) \n",
+						"⎢                                                                             \n",
+						"⎢                                                                   dv_dx     \n",
+						"⎢                                                                             \n",
+						"⎢                                                                   dw_dx     \n",
+						"⎢                                                                             \n",
+						"⎢                             ⎛ 2            2                                \n",
+						"⎢       -K⋅(P - Pₜ) - (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ - c⋅du_dy⋅ρ⋅v - c⋅du_dz\n",
+						"⎢       ──────────────────────────────────────────────────────────────────────\n",
+						"⎢                                                                 2⋅(c - u)   \n",
+						"⎢                                                                             \n",
+						"⎣                                                                   ds_dx     \n",
+						"\n",
+						"                 ⎞           ⎛   2                            ⎞⎤\n",
+						"dP_dy⋅v + dP_dz⋅w⎠ + (c - u)⋅⎝2⋅c ⋅drho_dx + c⋅du_dx⋅ρ - dP_dx⎠⎥\n",
+						"───────────────────────────────────────────────────────────────⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                        ⎞                                      ⎥\n",
+						"⋅ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c - u)⋅(c⋅du_dx⋅ρ + dP_dx)        ⎥\n",
+						"───────────────────────────────────────────────────────        ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                        ⎞                                      ⎥\n",
+						"⋅ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c - u)⋅(c⋅du_dx⋅ρ + dP_dx)        ⎥\n",
+						"───────────────────────────────────────────────────────        ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎥\n",
+						"                                                               ⎦"
+					]
+				},
+				"execution_count" : 17,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "dQ_dx_outflow_x1_upper = simplify(S * Inverse(lambda_waves) * L_outflow_x1_upper)\n", "dQ_dx_outflow_x1_upper" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 18,
+			"metadata" : {},
+			"outputs" : [ {
+				"name" : "stdout",
+				"output_type" : "stream",
+				"text" : [
+					"x0 = std::pow(c, 2);\n", "x1 = c - u;\n", "x2 = c*rho;\n", "x3 = du_dx*x2;\n", "x4 = rho*x0;\n",
+					"x5 = K*(P - P_t) + (beta - 1)*(dP_dy*v + dP_dz*w - du_dy*v*x2 - du_dz*w*x2 + dv_dy*x4 + dw_dz*x4);\n",
+					"x6 = (1.0/2.0)/x1;\n", "x7 = dP_dx + x3;\n", "dQ_dx[0] = x6*(x1*(-dP_dx + 2*drho_dx*x0 + x3) - x5)/x0;\n",
+					"dQ_dx[1] = x6*(x1*x7 + x5)/(c*rho);\n", "dQ_dx[2] = dv_dx;\n", "dQ_dx[3] = dw_dx;\n", "dQ_dx[4] = x6*(x1*x7 - x5);\n",
+					"dQ_dx[5] = ds_dx;\n"
+				]
+			} ],
+			"source" : [
+				"drho_dx = Symbol('drho_dx')\n", "du_dx = Symbol('du_dx')\n", "dv_dx = Symbol('dv_dx')\n", "dw_dx = Symbol('dw_dx')\n",
+				"dp_dx = Symbol('dP_dx')\n", "ds_dx = Symbol('ds_dx')\n", "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
+				"L = simplify(lambda_waves * Sinv * dQ_dx_def)\n", "\n", "T_phi = simplify(T_1[0] + T_2[0])\n",
+				"L_outflow_x1_upper = Matrix([K*(p - p_t) - (1 - beta)*T_phi, L[1], L[2], L[3], L[4], L[5]])\n",
+				"dQ_dx_outflow_x1_upper = simplify(S * Inverse(lambda_waves) * L_outflow_x1_upper)\n", "\n",
+				"common_exprs_upper, final_expr_upper = cse(dQ_dx_outflow_x1_upper, list=False)\n", "for var, val in common_exprs_upper:\n",
+				"    print(cxxcode(val, assign_to=var))\n", "print(cxxcode(final_expr_upper, assign_to='dQ_dx'))"
+			]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"### x1 Lower Outflow\n",
+				"We replace the $\\mathcal{L}_4$ component with a relaxation term for a specified pressure at the outflow boundary to obtain "
+				"$\\mathcal{\\tilde L}$, and then compute ${d\\tilde Q}/{dx} = S \\lambda^{-1} \\mathcal{\\tilde L}$."
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 19,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}\\left(c - u\\right) \\left(c du_{dx} \\rho - dP_{dx}\\right)\\\\u "
+					     "\\left(- c^{2} drho_{dx} + dP_{dx}\\right)\\\\dv_{dx} u\\\\dw_{dx} u\\\\K \\left(P - P_{t}\\right) - \\left(1 - "
+					     "\\beta\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + "
+					     "dP_{dy} v + dP_{dz} w\\right)\\\\ds_{dx} u\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                                 (c - u)⋅(c⋅du_dx⋅ρ - dP_dx)                 \n",
+						"⎢                                                                             \n",
+						"⎢                                     ⎛   2                ⎞                  \n",
+						"⎢                                   u⋅⎝- c ⋅drho_dx + dP_dx⎠                  \n",
+						"⎢                                                                             \n",
+						"⎢                                           dv_dx⋅u                           \n",
+						"⎢                                                                             \n",
+						"⎢                                           dw_dx⋅u                           \n",
+						"⎢                                                                             \n",
+						"⎢                     ⎛ 2            2                                        \n",
+						"⎢K⋅(P - Pₜ) - (1 - β)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅ρ⋅w + d\n",
+						"⎢                                                                             \n",
+						"⎣                                           ds_dx⋅u                           \n",
+						"\n",
+						"                 ⎤\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                 ⎥\n",
+						"                ⎞⎥\n",
+						"P_dy⋅v + dP_dz⋅w⎠⎥\n",
+						"                 ⎥\n",
+						"                 ⎦"
+					]
+				},
+				"execution_count" : 19,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"K = Symbol('K')\n", "p = Symbol('P')\n", "p_t = Symbol('P_t')\n", "beta = Symbol('beta')\n", "v = Symbol('v')\n",
+				"dp_dy = Symbol('dP_dy')\n", "du_dy = Symbol('du_dy')\n", "gamma = Symbol('gamma')\n", "dv_dy = Symbol('dv_dy')\n", "\n",
+				"T_phi = simplify(T_1[4] + T_2[4])\n",
+				"L_outflow_x1_lower = Matrix([L[0], L[1], L[2], L[3], K*(p - p_t) - (1 - beta)*T_phi, L[5]])\n", "L_outflow_x1_lower"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 20,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" :
+					    ["$\\displaystyle \\left[\\begin{matrix}\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} "
+					     "dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + "
+					     "\\left(c + u\\right) \\left(2 c^{2} drho_{dx} - c du_{dx} \\rho - dP_{dx}\\right)}{2 c^{2} \\left(c + "
+					     "u\\right)}\\\\\\frac{K \\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} "
+					     "dw_{dz} \\rho + c du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c + u\\right) "
+					     "\\left(c du_{dx} \\rho - dP_{dx}\\right)}{2 c \\rho \\left(c + u\\right)}\\\\dv_{dx}\\\\dw_{dx}\\\\\\frac{K "
+					     "\\left(P - P_{t}\\right) + \\left(\\beta - 1\\right) \\left(c^{2} dv_{dy} \\rho + c^{2} dw_{dz} \\rho + c "
+					     "du_{dy} \\rho v + c du_{dz} \\rho w + dP_{dy} v + dP_{dz} w\\right) + \\left(c + u\\right) \\left(- c du_{dx} "
+					     "\\rho + dP_{dx}\\right)}{2 \\left(c + u\\right)}\\\\ds_{dx}\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡                     ⎛ 2            2                                        \n",
+						"⎢K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅ρ⋅w + d\n",
+						"⎢─────────────────────────────────────────────────────────────────────────────\n",
+						"⎢                                                                   2         \n",
+						"⎢                                                                2⋅c ⋅(c + u) \n",
+						"⎢                                                                             \n",
+						"⎢                            ⎛ 2            2                                 \n",
+						"⎢       K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅\n",
+						"⎢       ──────────────────────────────────────────────────────────────────────\n",
+						"⎢                                                              2⋅c⋅ρ⋅(c + u)  \n",
+						"⎢                                                                             \n",
+						"⎢                                                                   dv_dx     \n",
+						"⎢                                                                             \n",
+						"⎢                                                                   dw_dx     \n",
+						"⎢                                                                             \n",
+						"⎢                            ⎛ 2            2                                 \n",
+						"⎢       K⋅(P - Pₜ) + (β - 1)⋅⎝c ⋅dv_dy⋅ρ + c ⋅dw_dz⋅ρ + c⋅du_dy⋅ρ⋅v + c⋅du_dz⋅\n",
+						"⎢       ──────────────────────────────────────────────────────────────────────\n",
+						"⎢                                                                 2⋅(c + u)   \n",
+						"⎢                                                                             \n",
+						"⎣                                                                   ds_dx     \n",
+						"\n",
+						"                ⎞           ⎛   2                            ⎞⎤\n",
+						"P_dy⋅v + dP_dz⋅w⎠ + (c + u)⋅⎝2⋅c ⋅drho_dx - c⋅du_dx⋅ρ - dP_dx⎠⎥\n",
+						"──────────────────────────────────────────────────────────────⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                       ⎞                                      ⎥\n",
+						"ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c + u)⋅(c⋅du_dx⋅ρ - dP_dx)        ⎥\n",
+						"──────────────────────────────────────────────────────        ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                       ⎞                                      ⎥\n",
+						"ρ⋅w + dP_dy⋅v + dP_dz⋅w⎠ + (c + u)⋅(-c⋅du_dx⋅ρ + dP_dx)       ⎥\n",
+						"───────────────────────────────────────────────────────       ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎥\n",
+						"                                                              ⎦"
+					]
+				},
+				"execution_count" : 20,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "dQ_dx_outflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_outflow_x1_lower)\n", "dQ_dx_outflow_x1_lower" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 21,
+			"metadata" : {},
+			"outputs" : [ {
+				"name" : "stdout",
+				"output_type" : "stream",
+				"text" : [
+					"x0 = std::pow(c, 2);\n", "x1 = c + u;\n", "x2 = c*rho;\n", "x3 = du_dx*x2;\n", "x4 = rho*x0;\n",
+					"x5 = K*(P - P_t) + (beta - 1)*(dP_dy*v + dP_dz*w + du_dy*v*x2 + du_dz*w*x2 + dv_dy*x4 + dw_dz*x4);\n",
+					"x6 = (1.0/2.0)/x1;\n", "x7 = -dP_dx + x3;\n", "dQ_dx[0] = x6*(x1*(-dP_dx + 2*drho_dx*x0 - x3) + x5)/x0;\n",
+					"dQ_dx[1] = x6*(x1*x7 + x5)/(c*rho);\n", "dQ_dx[2] = dv_dx;\n", "dQ_dx[3] = dw_dx;\n", "dQ_dx[4] = x6*(-x1*x7 + x5);\n",
+					"dQ_dx[5] = ds_dx;\n"
+				]
+			} ],
+			"source" : [
+				"drho_dx = Symbol('drho_dx')\n", "du_dx = Symbol('du_dx')\n", "dv_dx = Symbol('dv_dx')\n", "dw_dx = Symbol('dw_dx')\n",
+				"dp_dx = Symbol('dP_dx')\n", "ds_dx = Symbol('ds_dx')\n", "dQ_dx_def = Matrix([drho_dx, du_dx, dv_dx, dw_dx, dp_dx, ds_dx])\n",
+				"L = simplify(lambda_waves * Sinv * dQ_dx_def)\n", "\n", "T_phi = simplify(T_1[4] + T_2[4])\n",
+				"L_outflow_x1_lower = Matrix([L[0], L[1], L[2], L[3], K*(p - p_t) - (1 - beta)*T_phi, L[5]])\n",
+				"dQ_dx_outflow_x1_lower = simplify(S * Inverse(lambda_waves) * L_outflow_x1_lower)\n", "\n",
+				"common_exprs_lower, final_expr_lower = cse(dQ_dx_outflow_x1_lower, list=False)\n", "for var, val in common_exprs_lower:\n",
+				"    print(cxxcode(val, assign_to=var))\n", "print(cxxcode(final_expr_lower, assign_to='dQ_dx'))"
+			]
+		}
+	],
+	"metadata" : {
+		"kernelspec" : {"display_name" : "Python 3", "language" : "python", "name" : "python3"},
+		"language_info" : {
+			"codemirror_mode" : {"name" : "ipython", "version" : 3},
+			"file_extension" : ".py",
+			"mimetype" : "text/x-python",
+			"name" : "python",
+			"nbconvert_exporter" : "python",
+			"pygments_lexer" : "ipython3",
+			"version" : "3.11.5"
+		},
+		"orig_nbformat" : 4
+	},
+	"nbformat" : 4,
+	"nbformat_minor" : 2
 }
diff --git a/src/problems/RadDustMG/test_rad_dust_MG.cpp b/src/problems/RadDustMG/test_rad_dust_MG.cpp
index c5cc5ce63..17cba0312 100644
--- a/src/problems/RadDustMG/test_rad_dust_MG.cpp
+++ b/src/problems/RadDustMG/test_rad_dust_MG.cpp
@@ -70,8 +70,8 @@ template <> struct RadSystem_Traits<DustProblem> {
 
 template <>
 AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE auto
-RadSystem<DustProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> /*rad_boundaries*/, const double rho,
-							     const double /*Tgas*/) -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
+RadSystem<DustProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> /*rad_boundaries*/, const double rho, const double /*Tgas*/)
+    -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
 {
 	amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> exponents_and_values{};
 	for (int i = 0; i < nGroups_ + 1; ++i) {
@@ -82,9 +82,9 @@ RadSystem<DustProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<dou
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto
-RadSystem<DustProblem>::ComputeThermalRadiationMultiGroup(amrex::Real temperature,
-							  amrex::GpuArray<double, nGroups_ + 1> const &boundaries) -> quokka::valarray<amrex::Real, nGroups_>
+AMREX_GPU_HOST_DEVICE auto RadSystem<DustProblem>::ComputeThermalRadiationMultiGroup(amrex::Real temperature,
+										     amrex::GpuArray<double, nGroups_ + 1> const &boundaries)
+    -> quokka::valarray<amrex::Real, nGroups_>
 {
 	auto radEnergyFractions = ComputePlanckEnergyFractions(boundaries, temperature);
 	const double power = radiation_constant_ * temperature;
@@ -93,8 +93,9 @@ RadSystem<DustProblem>::ComputeThermalRadiationMultiGroup(amrex::Real temperatur
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto RadSystem<DustProblem>::ComputeThermalRadiationTempDerivativeMultiGroup(
-    amrex::Real temperature, amrex::GpuArray<double, nGroups_ + 1> const &boundaries) -> quokka::valarray<amrex::Real, nGroups_>
+AMREX_GPU_HOST_DEVICE auto RadSystem<DustProblem>::ComputeThermalRadiationTempDerivativeMultiGroup(amrex::Real temperature,
+												   amrex::GpuArray<double, nGroups_ + 1> const &boundaries)
+    -> quokka::valarray<amrex::Real, nGroups_>
 {
 	auto radEnergyFractions = ComputePlanckEnergyFractions(boundaries, temperature);
 	const double d_power_dt = radiation_constant_;
diff --git a/src/problems/RadMarshak/test_radiation_marshak.cpp b/src/problems/RadMarshak/test_radiation_marshak.cpp
index e7902309a..a2dcd9449 100644
--- a/src/problems/RadMarshak/test_radiation_marshak.cpp
+++ b/src/problems/RadMarshak/test_radiation_marshak.cpp
@@ -68,17 +68,17 @@ template <> AMREX_GPU_HOST_DEVICE auto RadSystem<SuOlsonProblem>::ComputeFluxMea
 
 static constexpr int nmscalars_ = Physics_Traits<SuOlsonProblem>::numMassScalars;
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<SuOlsonProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
-						 std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<SuOlsonProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
+									    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return std::pow(4.0 * Egas / alpha_SuOlson, 1. / 4.);
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<SuOlsonProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
-						 std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<SuOlsonProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
+									    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return (alpha_SuOlson / 4.0) * std::pow(Tgas, 4);
 }
diff --git a/src/problems/RadMarshakCGS/test_radiation_marshak_cgs.cpp b/src/problems/RadMarshakCGS/test_radiation_marshak_cgs.cpp
index fa4a1f997..b724c9ac4 100644
--- a/src/problems/RadMarshakCGS/test_radiation_marshak_cgs.cpp
+++ b/src/problems/RadMarshakCGS/test_radiation_marshak_cgs.cpp
@@ -70,17 +70,17 @@ template <> AMREX_GPU_HOST_DEVICE auto RadSystem<SuOlsonProblemCgs>::ComputeFlux
 
 static constexpr int nmscalars_ = Physics_Traits<SuOlsonProblemCgs>::numMassScalars;
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<SuOlsonProblemCgs>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
-						    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<SuOlsonProblemCgs>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
+									       std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return std::pow(4.0 * Egas / alpha_SuOlson, 1. / 4.);
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<SuOlsonProblemCgs>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
-						    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<SuOlsonProblemCgs>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
+									       std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return (alpha_SuOlson / 4.0) * std::pow(Tgas, 4);
 }
diff --git a/src/problems/RadMatterCoupling/test_radiation_matter_coupling.cpp b/src/problems/RadMatterCoupling/test_radiation_matter_coupling.cpp
index 43b171c77..ed17b1f2b 100644
--- a/src/problems/RadMatterCoupling/test_radiation_matter_coupling.cpp
+++ b/src/problems/RadMatterCoupling/test_radiation_matter_coupling.cpp
@@ -68,17 +68,17 @@ template <> AMREX_GPU_HOST_DEVICE auto RadSystem<CouplingProblem>::ComputeFluxMe
 
 static constexpr int nmscalars_ = Physics_Traits<CouplingProblem>::numMassScalars;
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<CouplingProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
-						  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<CouplingProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
+									     std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return std::pow(4.0 * Egas / alpha_SuOlson, 1. / 4.);
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<CouplingProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
-						  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<CouplingProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
+									     std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return (alpha_SuOlson / 4.0) * std::pow(Tgas, 4);
 }
diff --git a/src/problems/RadMatterCouplingRSLA/test_radiation_matter_coupling_rsla.cpp b/src/problems/RadMatterCouplingRSLA/test_radiation_matter_coupling_rsla.cpp
index 465fb7edf..e61548026 100644
--- a/src/problems/RadMatterCouplingRSLA/test_radiation_matter_coupling_rsla.cpp
+++ b/src/problems/RadMatterCouplingRSLA/test_radiation_matter_coupling_rsla.cpp
@@ -70,17 +70,17 @@ template <> AMREX_GPU_HOST_DEVICE auto RadSystem<CouplingProblem>::ComputeFluxMe
 
 static constexpr int nmscalars_ = Physics_Traits<CouplingProblem>::numMassScalars;
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<CouplingProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
-						  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<CouplingProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
+									     std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return std::pow(4.0 * Egas / alpha_SuOlson, 1. / 4.);
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<CouplingProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
-						  std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<CouplingProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
+									     std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return (alpha_SuOlson / 4.0) * std::pow(Tgas, 4);
 }
diff --git a/src/problems/RadSuOlson/test_radiation_SuOlson.cpp b/src/problems/RadSuOlson/test_radiation_SuOlson.cpp
index 4e6570cf9..a5cda31e0 100644
--- a/src/problems/RadSuOlson/test_radiation_SuOlson.cpp
+++ b/src/problems/RadSuOlson/test_radiation_SuOlson.cpp
@@ -83,17 +83,17 @@ template <> AMREX_GPU_HOST_DEVICE auto RadSystem<MarshakProblem>::ComputeFluxMea
 
 static constexpr int nmscalars_ = Physics_Traits<MarshakProblem>::numMassScalars;
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<MarshakProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
-						 std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<MarshakProblem>::ComputeTgasFromEint(const double /*rho*/, const double Egas,
+									    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return std::pow(4.0 * Egas / alpha_SuOlson, 1. / 4.);
 }
 
 template <>
-AMREX_GPU_HOST_DEVICE auto
-quokka::EOS<MarshakProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
-						 std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/) -> double
+AMREX_GPU_HOST_DEVICE auto quokka::EOS<MarshakProblem>::ComputeEintFromTgas(const double /*rho*/, const double Tgas,
+									    std::optional<amrex::GpuArray<amrex::Real, nmscalars_>> const & /*massScalars*/)
+    -> double
 {
 	return (alpha_SuOlson / 4.0) * (Tgas * Tgas * Tgas * Tgas);
 }
diff --git a/src/problems/RadhydroPulseMGint/test_radhydro_pulse_MG_int.cpp b/src/problems/RadhydroPulseMGint/test_radhydro_pulse_MG_int.cpp
index c08b26d59..619fee4dd 100644
--- a/src/problems/RadhydroPulseMGint/test_radhydro_pulse_MG_int.cpp
+++ b/src/problems/RadhydroPulseMGint/test_radhydro_pulse_MG_int.cpp
@@ -172,8 +172,8 @@ auto compute_kappa(const double nu, const double Tgas) -> double
 
 template <>
 AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE auto
-RadSystem<MGProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> const rad_boundaries, const double rho,
-							   const double Tgas) -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
+RadSystem<MGProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> const rad_boundaries, const double rho, const double Tgas)
+    -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
 {
 	amrex::GpuArray<double, nGroups_ + 1> exponents{};
 	amrex::GpuArray<double, nGroups_ + 1> kappa_lower{};
diff --git a/src/problems/RadhydroShockMultigroup/test_radhydro_shock_multigroup.cpp b/src/problems/RadhydroShockMultigroup/test_radhydro_shock_multigroup.cpp
index d36bd680f..a72799c2b 100644
--- a/src/problems/RadhydroShockMultigroup/test_radhydro_shock_multigroup.cpp
+++ b/src/problems/RadhydroShockMultigroup/test_radhydro_shock_multigroup.cpp
@@ -77,8 +77,8 @@ template <> struct quokka::EOS_Traits<ShockProblem> {
 
 template <>
 AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE auto
-RadSystem<ShockProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> /*rad_boundaries*/, const double rho,
-							      const double /*Tgas*/) -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
+RadSystem<ShockProblem>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> /*rad_boundaries*/, const double rho, const double /*Tgas*/)
+    -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
 {
 	amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> exponents_and_values{};
 	for (int i = 0; i < nGroups_ + 1; ++i) {
diff --git a/src/problems/ShockCloud/boundary_conditions.ipynb b/src/problems/ShockCloud/boundary_conditions.ipynb
index 2aefa6415..096838100 100644
--- a/src/problems/ShockCloud/boundary_conditions.ipynb
+++ b/src/problems/ShockCloud/boundary_conditions.ipynb
@@ -1,812 +1,1978 @@
 {
-	"cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Shock jump conditions"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "from sympy import *\n",
-    "init_printing(use_unicode=True)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The Euler equations in 1D are:\n",
-    "\n",
-    "$\\partial_t \\vec{U} + \\partial_x \\vec{F}(\\vec{U}) = 0$,\n",
-    "\n",
-    "where $\\vec{U} = [\\rho, u, P]$.\n",
-    "\n",
-    "Using the weak form of the Euler equations, and assuming a steady state on the left state $U_L$ and right side $U_R$ of the shock, we can obtain a jump discontinuity solution:\n",
-    "\n",
-    "$u_s (\\vec{U}_L - \\vec{U}_R) = {\\vec{F}(\\vec{U}_L) - \\vec{F}(\\vec{U}_R)}$\n",
-    "\n",
-    "where $u_s$ is the speed that the discontinuity travels (i.e., the shock speed).\n",
-    "\n",
-    "*A crucial simplification is obtained by working in the shock tube frame, where (by definition) $u_R = 0$.*"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}0\\\\p_{R}\\\\0\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡ 0 ⎤\n",
-       "⎢   ⎥\n",
-       "⎢p_R⎥\n",
-       "⎢   ⎥\n",
-       "⎣ 0 ⎦"
-      ]
-     },
-     "execution_count": 2,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "rho_R = Symbol('rho_R')\n",
-    "u_R = 0\n",
-    "p_R = Symbol('p_R')\n",
-    "gamma = Symbol('gamma')\n",
-    "e_R = p_R / (rho_R * (gamma - 1))\n",
-    "E_R = rho_R * (e_R + u_R**2 / 2)\n",
-    "U_R = Matrix([rho_R, rho_R*u_R, E_R])\n",
-    "\n",
-    "F_R = Matrix([rho_R*u_R,\n",
-    "        rho_R*u_R**2 + p_R,\n",
-    "        (E_R + p_R)*u_R])\n",
-    "F_R"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}\\rho_{L} u_{L}\\\\p_{L} + \\rho_{L} u_{L}^{2}\\\\u_{L} \\left(p_{L} + \\rho_{L} \\left(\\frac{p_{L}}{\\rho_{L} \\left(\\gamma - 1\\right)} + \\frac{u_{L}^{2}}{2}\\right)\\right)\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "⎡              ρ_L⋅u_L               ⎤\n",
-       "⎢                                    ⎥\n",
-       "⎢                        2           ⎥\n",
-       "⎢           p_L + ρ_L⋅u_L            ⎥\n",
-       "⎢                                    ⎥\n",
-       "⎢    ⎛          ⎛                 2⎞⎞⎥\n",
-       "⎢    ⎜          ⎜    p_L       u_L ⎟⎟⎥\n",
-       "⎢u_L⋅⎜p_L + ρ_L⋅⎜─────────── + ────⎟⎟⎥\n",
-       "⎣    ⎝          ⎝ρ_L⋅(γ - 1)    2  ⎠⎠⎦"
-      ]
-     },
-     "execution_count": 3,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "rho_L = Symbol('rho_L')\n",
-    "M = Symbol('M')\n",
-    "u_L = Symbol('u_L')\n",
-    "p_L = Symbol('p_L')\n",
-    "e_L = p_L / (rho_L * (gamma - 1))\n",
-    "E_L = rho_L * (e_L + u_L**2 / 2)\n",
-    "U_L = Matrix([rho_L, rho_L*u_L, E_L])\n",
-    "\n",
-    "F_L = Matrix([rho_L*u_L,\n",
-    "        rho_L*u_L**2 + p_L,\n",
-    "        (E_L + p_L)*u_L])\n",
-    "F_L"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, we want to obtain the left (post-shock) state $U_L$ in terms of the pre-shock state $U_R$ and the Mach number $\\mathcal{M}$, which is defined as $\\mathcal{M} = u_s / c_R$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "M = Symbol('M')\n",
-    "c_R = sqrt(gamma * p_R / rho_R)\n",
-    "u_s = M * c_R"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\rho_{L} = \\frac{M \\rho_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}{M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - u_{L}}$"
-      ],
-      "text/plain": [
-       "                 _______ \n",
-       "                ╱ γ⋅p_R  \n",
-       "       M⋅ρ_R⋅  ╱  ─────  \n",
-       "             ╲╱    ρ_R   \n",
-       "ρ_L = ───────────────────\n",
-       "            _______      \n",
-       "           ╱ γ⋅p_R       \n",
-       "      M⋅  ╱  ─────  - u_L\n",
-       "        ╲╱    ρ_R        "
-      ]
-     },
-     "execution_count": 5,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "rho_L_sol1 = solveset(Eq(u_s * (U_L[0] - U_R[0]), F_L[0] - F_R[0]), rho_L).args[0]\n",
-    "Eq(rho_L, rho_L_sol1)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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AAElFTkSuQmCC",
-      "text/latex": [
-       "$\\displaystyle \\rho_{L} = \\frac{p_{L} - p_{R}}{u_{L} \\left(M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - u_{L}\\right)}$"
-      ],
-      "text/plain": [
-       "              p_L - p_R        \n",
-       "ρ_L = ─────────────────────────\n",
-       "          ⎛      _______      ⎞\n",
-       "          ⎜     ╱ γ⋅p_R       ⎟\n",
-       "      u_L⋅⎜M⋅  ╱  ─────  - u_L⎟\n",
-       "          ⎝  ╲╱    ρ_R        ⎠"
-      ]
-     },
-     "execution_count": 6,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "rho_L_sol2 = solveset(Eq(u_s * (U_L[1] - U_R[1]), F_L[1] - F_R[1]), rho_L).args[0]\n",
-    "Eq(rho_L, rho_L_sol2)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 7,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\rho_{L} = \\frac{2 \\left(- M p_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + M p_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + \\gamma p_{L} u_{L}\\right)}{u_{L}^{2} \\left(M \\gamma \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - \\gamma u_{L} + u_{L}\\right)}$"
-      ],
-      "text/plain": [
-       "        ⎛            _______             _______            ⎞\n",
-       "        ⎜           ╱ γ⋅p_R             ╱ γ⋅p_R             ⎟\n",
-       "      2⋅⎜- M⋅p_L⋅  ╱  ─────  + M⋅p_R⋅  ╱  ─────  + γ⋅p_L⋅u_L⎟\n",
-       "        ⎝        ╲╱    ρ_R           ╲╱    ρ_R              ⎠\n",
-       "ρ_L = ───────────────────────────────────────────────────────\n",
-       "             ⎛        _______         _______              ⎞ \n",
-       "           2 ⎜       ╱ γ⋅p_R         ╱ γ⋅p_R               ⎟ \n",
-       "        u_L ⋅⎜M⋅γ⋅  ╱  ─────  - M⋅  ╱  ─────  - γ⋅u_L + u_L⎟ \n",
-       "             ⎝    ╲╱    ρ_R       ╲╱    ρ_R                ⎠ "
-      ]
-     },
-     "execution_count": 7,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "rho_L_sol3 = solveset(Eq(u_s * (U_L[2] - U_R[2]), F_L[2] - F_R[2]), rho_L).args[0]\n",
-    "Eq(rho_L, rho_L_sol3)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 8,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle u_{L} = \\frac{M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} \\left(\\rho_{L} - \\rho_{R}\\right)}{\\rho_{L}}$"
-      ],
-      "text/plain": [
-       "            _______            \n",
-       "           ╱ γ⋅p_R             \n",
-       "      M⋅  ╱  ───── ⋅(ρ_L - ρ_R)\n",
-       "        ╲╱    ρ_R              \n",
-       "u_L = ─────────────────────────\n",
-       "                 ρ_L           "
-      ]
-     },
-     "execution_count": 8,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "u_L_sol1 = solveset(Eq(u_s * (U_L[0] - U_R[0]), F_L[0] - F_R[0]), u_L).args[0]\n",
-    "Eq(u_L, u_L_sol1)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle u_{L} = \\frac{M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}{2} - \\frac{\\sqrt{\\frac{\\rho_{L} \\left(M^{2} \\gamma p_{R} \\rho_{L} - 4 \\rho_{R} \\left(p_{L} - p_{R}\\right)\\right)}{\\rho_{R}}}}{2 \\rho_{L}}$"
-      ],
-      "text/plain": [
-       "                           ________________________________________\n",
-       "            _______       ╱     ⎛ 2                              ⎞ \n",
-       "           ╱ γ⋅p_R       ╱  ρ_L⋅⎝M ⋅γ⋅p_R⋅ρ_L - 4⋅ρ_R⋅(p_L - p_R)⎠ \n",
-       "      M⋅  ╱  ─────      ╱   ────────────────────────────────────── \n",
-       "        ╲╱    ρ_R     ╲╱                     ρ_R                   \n",
-       "u_L = ───────────── - ─────────────────────────────────────────────\n",
-       "            2                             2⋅ρ_L                    "
-      ]
-     },
-     "execution_count": 9,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "u_L_sol2 = solveset(Eq(u_s * (U_L[1] - U_R[1]), F_L[1] - F_R[1]), u_L).args[0]\n",
-    "Eq(u_L, u_L_sol2)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
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-      "text/latex": [
-       "$\\displaystyle p_{L} = M \\rho_{L} u_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + p_{R} - \\rho_{L} u_{L}^{2}$"
-      ],
-      "text/plain": [
-       "                    _______                 \n",
-       "                   ╱ γ⋅p_R                 2\n",
-       "p_L = M⋅ρ_L⋅u_L⋅  ╱  ─────  + p_R - ρ_L⋅u_L \n",
-       "                ╲╱    ρ_R                   "
-      ]
-     },
-     "execution_count": 10,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "p_L_sol1 = solveset(Eq(u_s * (U_L[1] - U_R[1]), F_L[1] - F_R[1]), p_L).args[0]\n",
-    "Eq(p_L, p_L_sol1)\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle p_{L} = \\frac{- M \\gamma \\rho_{L} u_{L}^{2} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + 2 M p_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + M \\rho_{L} u_{L}^{2} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + \\gamma \\rho_{L} u_{L}^{3} - \\rho_{L} u_{L}^{3}}{2 \\left(M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - \\gamma u_{L}\\right)}$"
-      ],
-      "text/plain": [
-       "                         _______               _______                  ______\n",
-       "                   2    ╱ γ⋅p_R               ╱ γ⋅p_R             2    ╱ γ⋅p_R\n",
-       "      - M⋅γ⋅ρ_L⋅u_L ⋅  ╱  ─────  + 2⋅M⋅p_R⋅  ╱  ─────  + M⋅ρ_L⋅u_L ⋅  ╱  ─────\n",
-       "                     ╲╱    ρ_R             ╲╱    ρ_R                ╲╱    ρ_R \n",
-       "p_L = ────────────────────────────────────────────────────────────────────────\n",
-       "                                            ⎛      _______        ⎞           \n",
-       "                                            ⎜     ╱ γ⋅p_R         ⎟           \n",
-       "                                          2⋅⎜M⋅  ╱  ─────  - γ⋅u_L⎟           \n",
-       "                                            ⎝  ╲╱    ρ_R          ⎠           \n",
-       "\n",
-       "_                        \n",
-       "             3          3\n",
-       "  + γ⋅ρ_L⋅u_L  - ρ_L⋅u_L \n",
-       "                         \n",
-       "─────────────────────────\n",
-       "                         \n",
-       "                         \n",
-       "                         \n",
-       "                         "
-      ]
-     },
-     "execution_count": 11,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "p_L_sol2 = solveset(Eq(u_s * (U_L[2] - U_R[2]), F_L[2] - F_R[2]), p_L).args[0]\n",
-    "Eq(p_L, p_L_sol2)\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 12,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle u_{L} = \\frac{p_{L} - p_{R}}{M \\rho_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}$"
-      ],
-      "text/plain": [
-       "          p_L - p_R    \n",
-       "u_L = ─────────────────\n",
-       "                _______\n",
-       "               ╱ γ⋅p_R \n",
-       "      M⋅ρ_R⋅  ╱  ───── \n",
-       "            ╲╱    ρ_R  "
-      ]
-     },
-     "execution_count": 12,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# write u_L in terms of U_R and p_L\n",
-    "u_L_sol_new1 = solveset(Eq(rho_L_sol1, rho_L_sol2), u_L).args[0].args[0]\n",
-    "Eq(u_L, u_L_sol_new1)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 13,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle p_{L} = \\frac{M^{2} \\gamma p_{R} \\rho_{L} - M^{2} \\gamma p_{R} \\rho_{R} + p_{R} \\rho_{L}}{\\rho_{L}}$"
-      ],
-      "text/plain": [
-       "       2              2                    \n",
-       "      M ⋅γ⋅p_R⋅ρ_L - M ⋅γ⋅p_R⋅ρ_R + p_R⋅ρ_L\n",
-       "p_L = ─────────────────────────────────────\n",
-       "                       ρ_L                 "
-      ]
-     },
-     "execution_count": 13,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# write p_L in terms of U_R and rho_L\n",
-    "p_L_sol_new1 = solveset(Eq(u_L_sol1, u_L_sol2), p_L).args[2].args[0]\n",
-    "Eq(p_L, p_L_sol_new1)\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 14,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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r41sFH0XkzZgYQ6l/mjSqF4IF2CPEoVVL46OJg0X+tCPn1xAu/zAeguPCP6oL/RH6RHbwQQtavfWpsEn4Nm9yLHNSPiR/5XvNvGg+vO+jxc9f4rN8LctZ+yrsc4yvY8bWWJxurgK9M1RCwKAdQTBB3dlHnDcJ2jKZCEnEY/AkeiG/sFjKxB+Bp3ZeJ8RXnENMywRVqw9+vFmxYCAkpAWa+qeFnYn3lZ6cqM9OcRAuwmIpk4WTuGjeWgui3CxCaPta/nmGKT+ZaO9C/mV4j5vFLdW1J0rjPRXuY/FsKiALWsxSaE1932q32jUU07y//qb8y/5ioS/95BX6MJVB/9d4JMQ78eeZ0n89Jg/VpcRoF/06/mPyHRNX5dEfc/JjqI7K4aWOccQtV86vYc9pKr7N85yDD8n/mnlxqXzImsIYLukWD/FLbV5lXekbz6uZv0pAxriFz9S45sXPMb6OHlt5xfbZNyGoqeMRethuCVuV+wApwpiYg4ZFaRlEuTYgDdBG01KkPehUnkkAa8WVP2fRPteDoNBoMPAnosy+smkfacsF5Yn8a/W8k/9jPXsp5jfJFeu9BX0KPAp31XMUnqm4iOcjud8mv2gmnGsC50FMUz9k+act8538kqBf65dUDfpzX3iKN9pU+bSXepS80uSlsI+N44wWlftgX3GxzpPyo/LkZWYns+zr9/KGD3jy8S9nh47i25SLyqaMyfmQ/JV36OesjKvgRdWH9jb1w16S4lwrH9YEMdrCWhGOpJRt2edWurXNX/ua2xsmHCbFNRWkfGcZX8r31LGVqthr3vSGrCRAIDJo+Eub5i1G9qApkZkLQUzUrS1KuQlnwKXJpIkv/yQUlRO7giajcGmhkhtlI8UHBsnCYcRmAi78n2fundISFyFtlEYlz2MKu+pxTtz78IQ3eNMt8eRNH5zzfk/NBr+hmJJ/2V/w5U557+MfyqhOWqQ9kdDwvdmXh+q2V2Dal3aBYWHsq801zVmnOYo3B9/OzYe049p40XzY4a5ej8XMXxofae5inuMFZ46jG71A9QTMPb5Gj62hON30NGgV3nEyfSKz3JaBeUqBBsYq35hZoPF/oTzKtx38Z9F2KN9E3OSsUads1Q+tIYJGq63yT5obBE7iIHSiJWTb89DWrKLNTufEfQyeLMQ8Hc3NEZjSB6VAhrAd+Cf2y1uZjaAoO2VDo3hM6TgvxqRY8nLILPthUplLCMyKWYwV7Gvn63iZgcp+mINva+N6Sj6kHWfhRfMhUE9OS5q/znF0YyzAc4+vY8bWIJxuxrZ0KfE1UQAaB6UBgj3vnJ7KrxS8uOlX+nEQPL0RNOkVj8kTgaf3fFoT+TRLR+OhsimX8u9S1vKjrQhoX8veLPYxHOZEmxPaFtODSxkvRj+7cU7c+/AEv9vU8gwj6lYKWEQbiykCf54//YcAkF4W0PiW/UEZbGPXtHkK6pLiwqtcJkFIOySoIbgPzrtb2up8Wi84tC7iyXiDD8r+mZRvlT/lzM2HNGt2Xoy4mQ9Be1pa0vxFy5uX0WlhGJ/bmcbXMWNrEE6rFdTUeoQoJj60SCU1i686EKGFhZOFi+0PDtenBZR43MIMC5pMFk+0EMSHCIMZ5/gOFhcCmnqG0u5/qANE2UyGEG9a3LirLbytwaI4LP4IrrS7szjJ7yykOpwb9zF4onHcdxliLKbg/FptRtv1Hz30E5o6vh1GH3YmYPm1ypA7kOKzoD+W2dLwyJ14Ep5JPHKfqPhV3M7LRxFlc05hwn/88gKXv9Qxf4SbrAkQhc/Ft6nP8nE9NR/SjFl5UfiYDxOzTGsubf6i9Wm9nBaJ43I7x/g6ZmwNwunmuDZffypNGExyB0nx2AKqksJYDJsFsXRXE03neduTFQs456lYkGuCXJkMBn1eeD6Ru9QQFFHmdV4A90N4IjyVGtU+EEZhqraCdY3Pan6pTMqobU2+lP+rFCkzEcC5pYhwgQAYPu+ShefWZ3LU8sjjbM4uzFrjvQaA4vT2WZm+dNfyy/zSuJ6NDylLdZqbF82HWadOaF3U/CU+SwJ7s35OiMUxWc0+vo4ZW0NxWq2gdkxPXksadR6LdE07RhUJO7StRbyd8mHRRgNTDhb87/QQJ5Qls688oi2aUht7GjEYT9Irr9kxVRnVSU7+aMLQ/HaEbPml/kt9jTDWJ8jvE+Jo5iBSmfAWgiNaQoT/V/LrK1PBpj0IXB0fUlf15yheVHzz4f1LcGeM7un7vUHCNMzRPZGukm9UV+qF0J74oaf6o7zB9FhcF43TzSiYHPlcCCD9d86/ielZGBEU3h2qiOKypUY+EF9NZhFNTI42ha24sHUqMy3uIfIKf07GE0wiXrNiGstAyII4B4VJv3OmjUUzlS9rlxQ/3V5lYuqQwvGfqr/5lhnb/pRJ3XC3LqjIn+1e+IxtAYhtetrDFl/iR/w3S8IBPAaNa0BS/LOM7VjOUbyotJvlQ7V9qvGVxsRi5q9UYZnUuYODsAlz2TEYHZOG+sQyr258UTfRIJwsqN2DdW2/aDzSwhbqJjfaFLQXEIIX34VLZ+nufbNfhbGNV93KUxhaD85IbYU6eMaGI/xAnQnl3rv9ew5M95XRrs1eF+3hKj/nWkphiK07zlmdRMoXYYH/n00aNN6cKa8skxeOp/JveDGmfS3/3m1EhW2Jro4PAT/2WdNvR3SI+fAI0CpJFjN/ZXXnhXDM0Y0s6eTWqxxfsZWDcLKgNjlPTJJhh8E1aQ65yTdJ4SvMpINnbON7mRwWL4WZpUPA5QQEeyaBcpscoakPjzHtJo88nzAZVrBE01YKwlUNwZjCVxbXfHh8h26BD/NxliN1lXyjOWDUdnneoJnsi8fJgtpMnHFKtmL0Kff1T6nKKtL24RmFilKIWEObU5vYtmoEtTiBJg3Y0e1UPmmrjgkwEdqxpqzkKZOza3dKg+B4q4dLPtysrsVV0PZIWPCikPpsTQCkNpkPT+hV8Ud1PbhGvlGd0LQftV1+AkR7k64BJwtqe7vYgUZgeQgwMelBIEOjlhMTaO1TIHmcIfaQb5wAdzI5/4FfbTsd/78QN8b7Qe78ExhymtaIQOxz8+EaO7enTerz3iM3PUk26T0Wp4ebRMmNNgLrRwCBjPNiaRuCFnNubQpNFluXfArk73r4TiFbM/xBeWsLWW40b7vkLxPNAE9eJ6KY1ouA+XC9feuWnQmBmzOV42KMgBE4LwL5thM38NB6VbdQjqgW59G4RXxI6EOb9r7In7R8560lxBVx7FwPAubD9fSlW3IhBKxRuxDwLtYIzImABCG2nNBwISxBaL2m2nJEI5YWYPLukMrnTBq3lLmdnGvQ3soPjdwL+bc0cPIzrQwB8+HKOtTNuQgCDz5+/HiRgl2oETAC8yKgRZLPcCAw8W0zvnFW+y9YeQ8n5YlmjssACFsmI3AQAfPhQYgcwQjsRcCC2l54HGgElouAFkiENIQ1DvjyLSYLV8vtzsXW3Hy42K5zxa8EAW99XklHuBpGYAYE0vYkV+ZP/sjtDPVzlttAwHy4jX52K2dCwILaTMA6WyNwaQSkyeAMWPpuGmfDTEbg7AiYD88OuQtcGQK+9bmyDnVzjECBQPhuWlwsiyA7jcDZEDAfng1qF7Q2BCyora1H3R4j0EaA/4NNW0/tELuMwPkQMB+eD2uXtDIE/g92fUQ5lvt0bgAAAABJRU5ErkJggg==",
-      "text/latex": [
-       "$\\displaystyle \\rho_{L} = \\frac{2 \\gamma p_{R} \\rho_{R}}{2 M^{2} \\gamma p_{R} - M \\gamma \\rho_{R} u_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - 3 M \\rho_{R} u_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + \\gamma \\rho_{R} u_{L}^{2} + \\rho_{R} u_{L}^{2}}$"
-      ],
-      "text/plain": [
-       "                                           2⋅γ⋅p_R⋅ρ_R                        \n",
-       "ρ_L = ────────────────────────────────────────────────────────────────────────\n",
-       "                                   _______                   _______          \n",
-       "         2                        ╱ γ⋅p_R                   ╱ γ⋅p_R           \n",
-       "      2⋅M ⋅γ⋅p_R - M⋅γ⋅ρ_R⋅u_L⋅  ╱  ─────  - 3⋅M⋅ρ_R⋅u_L⋅  ╱  ─────  + γ⋅ρ_R⋅u\n",
-       "                               ╲╱    ρ_R                 ╲╱    ρ_R            \n",
-       "\n",
-       "              \n",
-       "──────────────\n",
-       "              \n",
-       "  2          2\n",
-       "_L  + ρ_R⋅u_L \n",
-       "              "
-      ]
-     },
-     "execution_count": 14,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# write rho_L in terms of U_R and u_L\n",
-    "rho_L_sol_new1 = solveset(Eq(p_L_sol1, p_L_sol2), rho_L).args[0]\n",
-    "Eq(rho_L, rho_L_sol_new1)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We finally obtain:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 15,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle p_{L} = \\frac{p_{R} \\left(2 M^{2} \\gamma - \\gamma + 1\\right)}{\\gamma + 1}$"
-      ],
-      "text/plain": [
-       "          ⎛   2          ⎞\n",
-       "      p_R⋅⎝2⋅M ⋅γ - γ + 1⎠\n",
-       "p_L = ────────────────────\n",
-       "             γ + 1        "
-      ]
-     },
-     "execution_count": 15,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# substitute everything\n",
-    "p_L_sol_final = simplify(solveset(Eq(p_L, p_L_sol_new1.subs(rho_L, rho_L_sol_new1).subs(u_L, u_L_sol_new1)), p_L).args[0])\n",
-    "p_L_lambda = lambdify([M, p_R, gamma], p_L_sol_final)\n",
-    "Eq(p_L, p_L_sol_final)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 16,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle u_{L} = \\frac{2 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} \\left(M^{2} - 1\\right)}{M \\left(\\gamma + 1\\right)}$"
-      ],
-      "text/plain": [
-       "            _______         \n",
-       "           ╱ γ⋅p_R  ⎛ 2    ⎞\n",
-       "      2⋅  ╱  ───── ⋅⎝M  - 1⎠\n",
-       "        ╲╱    ρ_R           \n",
-       "u_L = ──────────────────────\n",
-       "            M⋅(γ + 1)       "
-      ]
-     },
-     "execution_count": 16,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# back substitute\n",
-    "u_L_sol_final = simplify(u_L_sol_new1.subs(p_L, p_L_sol_final))\n",
-    "u_L_lambda = lambdify([M, rho_R, p_R, gamma], u_L_sol_final)\n",
-    "Eq(u_L, u_L_sol_final)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
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-      "text/latex": [
-       "$\\displaystyle \\rho_{L} = \\frac{M^{2} \\rho_{R} \\left(\\gamma + 1\\right)}{M^{2} \\gamma - M^{2} + 2}$"
-      ],
-      "text/plain": [
-       "       2            \n",
-       "      M ⋅ρ_R⋅(γ + 1)\n",
-       "ρ_L = ──────────────\n",
-       "       2      2     \n",
-       "      M ⋅γ - M  + 2 "
-      ]
-     },
-     "execution_count": 17,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# back substitute\n",
-    "rho_L_sol_final = simplify(rho_L_sol_new1.subs(u_L, u_L_sol_final))\n",
-    "rho_L_lambda = lambdify([M, rho_R, gamma], rho_L_sol_final)\n",
-    "Eq(rho_L, rho_L_sol_final)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 65,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "const Real x0 = std::pow(M, 2);\n",
-      "const Real x1 = gamma + 1;\n",
-      "const Real x2 = gamma*x0;\n",
-      "const Real x3 = 1.0/x1;\n",
-      "const Real x4 = std::sqrt(gamma*p_R/rho_R);\n",
-      "const Real rho_L = rho_R*x0*x1/(-x0 + x2 + 2);\n",
-      "const Real u_L = 2*x3*x4*(x0 - 1)/M;\n",
-      "const Real P_L = p_R*x3*(-gamma + 2*x2 + 1);\n",
-      "const Real v_shock = M*x4;\n"
-     ]
-    }
-   ],
-   "source": [
-    "## cxxcode\n",
-    "vars = ['rho_L', 'u_L', 'P_L', 'v_shock']\n",
-    "exprs = [rho_L_sol_final, u_L_sol_final, p_L_sol_final, u_s]\n",
-    "\n",
-    "optim_exprs = cse(exprs)\n",
-    "expr1 = optim_exprs[0]\n",
-    "expr2 = optim_exprs[1]\n",
-    "for var, expr in expr1:\n",
-    "    print(\"const Real \" + cxxcode(expr, assign_to=var))\n",
-    "for var, expr in zip(vars, expr2):\n",
-    "    print(\"const Real \" + cxxcode(expr, assign_to=var))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Converting between parameterizations\n",
-    "\n",
-    "We want convert between (pre-shock, right-side) background density $\\rho_R$, background pressure $p_R$ and wind velocity $v_{\\text{wind}}$ and Mach number $\\mathcal{M}$.\n",
-    "\n",
-    "That is:\n",
-    "\n",
-    "$\\rho_{bg} = \\rho_{R}$,\n",
-    "\n",
-    "$p_{bg} = p_R$,\n",
-    "\n",
-    "and\n",
-    "\n",
-    "$v_{\\text{wind}} = u_L$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 18,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle \\left\\{\\frac{u_{L} \\left(\\gamma + 1\\right)}{4 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}} - \\frac{\\sqrt{\\frac{16 \\gamma p_{R}}{\\rho_{R}} + u_{L}^{2} \\left(\\gamma + 1\\right)^{2}}}{4 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}, \\frac{u_{L} \\left(\\gamma + 1\\right)}{4 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}} + \\frac{\\sqrt{\\frac{16 \\gamma p_{R}}{\\rho_{R}} + u_{L}^{2} \\left(\\gamma + 1\\right)^{2}}}{4 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}\\right\\} \\setminus \\left\\{0\\right\\}$"
-      ],
-      "text/plain": [
-       "⎧                    __________________________                      _________\n",
-       "⎪                   ╱ 16⋅γ⋅p_R      2        2                      ╱ 16⋅γ⋅p_R\n",
-       "⎪                  ╱  ──────── + u_L ⋅(γ + 1)                      ╱  ────────\n",
-       "⎪ u_L⋅(γ + 1)    ╲╱     ρ_R                       u_L⋅(γ + 1)    ╲╱     ρ_R   \n",
-       "⎨───────────── - ──────────────────────────────, ───────────── + ─────────────\n",
-       "⎪      _______                 _______                 _______                \n",
-       "⎪     ╱ γ⋅p_R                 ╱ γ⋅p_R                 ╱ γ⋅p_R                 \n",
-       "⎪4⋅  ╱  ─────            4⋅  ╱  ─────            4⋅  ╱  ─────            4⋅  ╱\n",
-       "⎩  ╲╱    ρ_R               ╲╱    ρ_R               ╲╱    ρ_R               ╲╱ \n",
-       "\n",
-       "_________________⎫      \n",
-       "      2        2 ⎪      \n",
-       " + u_L ⋅(γ + 1)  ⎪      \n",
-       "                 ⎪      \n",
-       "─────────────────⎬ \\ {0}\n",
-       " _______         ⎪      \n",
-       "╱ γ⋅p_R          ⎪      \n",
-       "  ─────          ⎪      \n",
-       "   ρ_R           ⎭      "
-      ]
-     },
-     "execution_count": 18,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "v_wind = Symbol(r'v_{\\text{wind}}')\n",
-    "rho_bg = Symbol(r'\\rho_{bg}')\n",
-    "p_bg = Symbol(r'p_{bg}')\n",
-    "Mach_solve = solveset(Eq(u_L, u_L_sol_final), M)\n",
-    "Mach_solve"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 19,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/latex": [
-       "$\\displaystyle M = \\frac{v_{\\text{wind}} \\left(\\gamma + 1\\right)}{4 \\sqrt{\\frac{\\gamma p_{bg}}{\\rho_{bg}}}} + \\frac{\\sqrt{v_{\\text{wind}}^{2} \\left(\\gamma + 1\\right)^{2} + \\frac{16 \\gamma p_{bg}}{\\rho_{bg}}}}{4 \\sqrt{\\frac{\\gamma p_{bg}}{\\rho_{bg}}}}$"
-      ],
-      "text/plain": [
-       "                                  _________________________________________\n",
-       "                                 ╱                2        2   16⋅γ⋅p_{bg} \n",
-       "                                ╱  v_{\\text{wind}} ⋅(γ + 1)  + ─────────── \n",
-       "    v_{\\text{wind}}⋅(γ + 1)   ╲╱                                \\rho_{bg}  \n",
-       "M = ─────────────────────── + ─────────────────────────────────────────────\n",
-       "             ___________                          ___________              \n",
-       "            ╱  γ⋅p_{bg}                          ╱  γ⋅p_{bg}               \n",
-       "       4⋅  ╱  ─────────                     4⋅  ╱  ─────────               \n",
-       "         ╲╱   \\rho_{bg}                       ╲╱   \\rho_{bg}               "
-      ]
-     },
-     "execution_count": 19,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# use positive root (only physical solution)\n",
-    "Mach_sol = Mach_solve.args[0].args[1].subs(u_L, v_wind).subs(rho_R, rho_bg).subs(p_R, p_bg)\n",
-    "compute_mach_number = lambdify([v_wind, rho_bg, p_bg, gamma], Mach_sol)\n",
-    "Eq(M, Mach_sol)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 28,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle M = 1.42225795167899$"
-      ],
-      "text/plain": [
-       "M = 1.42225795167899"
-      ]
-     },
-     "execution_count": 28,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# numerical example\n",
-    "\n",
-    "import numpy as np\n",
-    "mH = 1.673533e-24 # H mass (g)\n",
-    "kB = 1.380649e-16 # Boltzmann constant (cgs)\n",
-    "gamma_bg = 5./3.  # assume gamma is constant\n",
-    "\n",
-    "v_wind_kms = 200. # km/s\n",
-    "nH_bg = 1.0e-3 # cm^-3\n",
-    "rho_bg = nH_bg * mH # mu = 1 for OOM estimate\n",
-    "P_bg = 1.0e4 # K cm^-3\n",
-    "\n",
-    "# compute Mach number of shock\n",
-    "Mach_shock = compute_mach_number(v_wind_kms * 1.0e5, nH_bg * mH, P_bg * kB, 5./3.)\n",
-    "Eq(M, Mach_shock)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 24,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGIAAAARCAYAAAAmE3lhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADZUlEQVRYCe2Y61EUQRCAD4oAKMgAMlCIQMgANQIkAy3/8Y/CDJAIVDKQDBAykAykyOD8vmV7q3duuZs7C7Wu6Kq5fmz3TE8/ZvZ2ZTwej5YRjo+Pt9jXh3ZvO+A7eeQ3raxDyE5b5hd4m3GK7LZTaIlavdIu+Gn2K8uYCDZsEs7A+0UQ3sPvI79M8mvoE2QXysDrIGXqdcmArtJzjiGYZb86ZLQEMiv8KO+DQNgd94xvIUf2Dnod3CRBObQ68mfyArIqvQftyd8a+2VNxB7h+EkArO4MdoKBt2OE14yJowrZFWMv2dfqYTYIM+2XNREG/JZAWt1DEAkyYd4dJcSR5HOhVu9Be/J3pv2aNm3mbT8vqmv4z8oD4G3nQ/BjGwvV/wLjpxU4BC8U8vyGEckY0gvZRq1eGJS41r5JBMYfMfCN4gD6nNElApkJOgA/tjkeLwbM6TlstcwDBnFuX7AxCR5J8Sa10S46rbhMVq1eO90EqrJfax30TBR8yyhbVdnQOTrC1iB6KZpAL7grZJ/AVYBu70KtMlpcya6+mMc/9Dcrl6vVe2y6TTvCszQC/Qb+pNA22KWsUcHukmGlebHNXaXFOk/G4pud5z6zj2XB5fWjiv1fUauX7TNdZb+Kc01rgg24rZiPJVtZWffeDV2CHTPtean/V3n25dHqWd/9p9AB+GbfkO6vhJDlCz9kWTdkcbnnZw1du44dEWC15IWVe+TodHSMbAkmcKEjhnmf9I5gfv3fBnedAG1xjcAGzwJqeGUJoiOiwGr10hQ9cqZ9ToQOlQHvqh3Hrayv4PtYAtpjSQiHH7jKX+wXSmDN9K1vu+C4nMPM5ETXe2/E5414Ln7J8KUg9lqrl+fI9Ez71aTda692Izs61OpYWeFYmNkN98h7tvHwX2H8sajcvH/e/NTRDWRH8M0+wCbkDtx0PvQIeh3kXXkoLyCr1XO9McPPIR3U2OeOsHLOMbJCvKQMrpXhBzC/0XxhlNB1TH6AvpvZAS/UKXmuBenv2JkMu7iEKKyQxx53Ebhv8St8n1sPmyjKHzF5wlPX+aOPfizsp1srLFq9WRfeZPohreyg5NczmSOQj6Ysn0kT5MH7AbltvvWchJkh7Cnko6n3YBpDkD2q3rY6doRkcxyBTVDvVdGHzzA9Ar8BRWCJ/3OY4W0AAAAASUVORK5CYII=",
-      "text/latex": [
-       "$\\displaystyle u_{L} = 200.0$"
-      ],
-      "text/plain": [
-       "u_L = 200.0"
-      ]
-     },
-     "execution_count": 24,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute post-shock fluid velocity in km/s\n",
-    "Eq(u_L, u_L_lambda(Mach_shock, nH_bg * mH, P_bg * kB, 5./3.) / 1.0e5)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 26,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": "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",
-      "text/latex": [
-       "$\\displaystyle u_{s} = 527.384360142785$"
-      ],
-      "text/plain": [
-       "uₛ = 527.384360142785"
-      ]
-     },
-     "execution_count": 26,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "# compute shock velocity in km/s\n",
-    "u_s_lambda = lambdify([M, rho_R, p_R, gamma], u_s)\n",
-    "Eq(Symbol('u_s'), u_s_lambda(Mach_shock, nH_bg * mH, P_bg * kB, 5./3.) / 1.0e5)"
-   ]
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.11.5"
-  },
-  "orig_nbformat": 4
- },
- "nbformat": 4,
- "nbformat_minor": 2
+	"cells" : [
+		{"cell_type" : "markdown", "metadata" : {}, "source" : ["# Shock jump conditions"]},
+		{
+			"cell_type" : "code",
+			"execution_count" : 1,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [ "from sympy import *\n", "init_printing(use_unicode=True)" ]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"The Euler equations in 1D are:\n", "\n", "$\\partial_t \\vec{U} + \\partial_x \\vec{F}(\\vec{U}) = 0$,\n", "\n",
+				"where $\\vec{U} = [\\rho, u, P]$.\n", "\n",
+				"Using the weak form of the Euler equations, and assuming a steady state on the left state $U_L$ and right side $U_R$ of the "
+				"shock, we can obtain a jump discontinuity solution:\n",
+				"\n", "$u_s (\\vec{U}_L - \\vec{U}_R) = {\\vec{F}(\\vec{U}_L) - \\vec{F}(\\vec{U}_R)}$\n", "\n",
+				"where $u_s$ is the speed that the discontinuity travels (i.e., the shock speed).\n", "\n",
+				"*A crucial simplification is obtained by working in the shock tube frame, where (by definition) $u_R = 0$.*"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 2,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
+					    "iVBORw0KGgoAAAANSUhEUgAAACgAAABLCAYAAAACnsWZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEBUlEQVRoBe2a3U0cMRSFF8RzRDZSCth0QJIKAh"
+					    "3kpwJCB0E8wRsiHZAOktABSQUIOoACIgWhVJDzzfrOeAfPjNeehSjylcz1z/W9Z47tWdvD2uHh4dZkMrlUCsnZ0dHRu1DDWHXyfy1fs5A/"
+					    "ta1teA2flcfYlxu/"
+					    "sKL8ScDvjureUu8DPBXihwC0gEcxvyxUqKA6qu4BpDJZ5NSY+C0nL5ROVJf9wOvJiLyOAsIcvpDeV2Kq7CudKx+"
+					    "cW17XwWw2QIH4qCib0mcWTfk75SmfWl2qzgaowKzyqwCAC9VtC+xmoC26agyA24p2G4ho84/2ZMkCGMnONBmdOmYBVH8Lzpzrkkcf4i5gVv/"
+					    "MMik6l8HQ3DMcxi7vxWTJAuheJwQPDaPV2WJJApkF0EX8IT0LRDcGaU+WMQB+V/RXAQQvVXflsRwwGa7KBigA/NjfSlc/"
+					    "7oRUnuF9r7RLOUf83UyOH9hic/"
+					    "BamkWBfqNy6BdGTfEyCkABuVPIvfiw8ZbZQxwfKs2yAEzjrelVGGy4SMsVBtN4a3oVBhsu0nKFwTTeml6FwYaLtFxhMI23plfUflD7vS114YaAjSi7"
+					    "5Jkrc6Qkv+v2hMqOK7FD/"
+					    "EEAuLVCfirNKCtxi8WpjXPJSmSQQYGAPS6CENi6UV19k6Uyt7KflGpRO2wDHs2ZhR035xROevSnLUpiGPQBAfZryzOBJwpaaZfnqHmuRN89ADnNTRg"
+					    "3XtHXcgB8ooSYnpfcXznj6SfSsIG0z7ncJ9+ZXWUx/8N8bdvSwm1E6JhKm8lzywDwjyuYtra2BgiMVIC9RoB/"
+					    "88qWpR4Wa3EPCbih4+gv67RhmQhNwAVGFJDbVQAvzCnVz1THkLOYqhtY5bm3vlT5qXS0RAGUU4Ix/xieShwIgHH+DbEK29XKd/"
+					    "1Z6W27ubOev1EA1R/"
+					    "2ECa7rVjegTsqhy6HmA4127JhjrIwALnAtsq9EguQgNyzXKF7Pc4beaD2PGPRLM0giyRGCFgz0tdBD2Hzr21PfTVFZMOrhvKgrA9ZyBHzD2f2su7sI"
+					    "luG395xB66v2R8rw1BjsyUdmhpmW+veIZYTbqwOnDUBp0r3Pl2ZN7WxKKqFYXWm1cbU4JJpKRkCyE+a/7O2lPMxjAeHeIwgOT4KwBz26FsY/"
+					    "O8Z7H3NLPP0es+VL+7LEFbbirnyxb1mIzHDTqfapbT622aA9mTJeg9qeNnpDMl0yKCvPQugHFvwvo1ozEN0YswF2OnYa+BokCy5AENzz8AYu+"
+					    "WLuzHSpTl7zAKNxmD7bBIw7a7KHWI8c5QMXWWwvS9f3GEoRmCrfHGPYerBbcZYJCsFXQDm0vvPM+ifSa61fWo/"
+					    "8KP+ozdgAMjGsuufcmzTie2qxA5bQf9/AVFzKbuXGLjzAAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}0\\\\p_{R}\\\\0\\end{matrix}\\right]$"],
+					"text/plain" : [ "⎡ 0 ⎤\n", "⎢   ⎥\n", "⎢p_R⎥\n", "⎢   ⎥\n", "⎣ 0 ⎦" ]
+				},
+				"execution_count" : 2,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"rho_R = Symbol('rho_R')\n", "u_R = 0\n", "p_R = Symbol('p_R')\n", "gamma = Symbol('gamma')\n",
+				"e_R = p_R / (rho_R * (gamma - 1))\n", "E_R = rho_R * (e_R + u_R**2 / 2)\n", "U_R = Matrix([rho_R, rho_R*u_R, E_R])\n", "\n",
+				"F_R = Matrix([rho_R*u_R,\n", "        rho_R*u_R**2 + p_R,\n", "        (E_R + p_R)*u_R])\n", "F_R"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 3,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" : ["$\\displaystyle \\left[\\begin{matrix}\\rho_{L} u_{L}\\\\p_{L} + \\rho_{L} u_{L}^{2}\\\\u_{L} "
+							"\\left(p_{L} + \\rho_{L} \\left(\\frac{p_{L}}{\\rho_{L} \\left(\\gamma - 1\\right)} + "
+							"\\frac{u_{L}^{2}}{2}\\right)\\right)\\end{matrix}\\right]$"],
+					"text/plain" : [
+						"⎡              ρ_L⋅u_L               ⎤\n", "⎢                                    ⎥\n",
+						"⎢                        2           ⎥\n", "⎢           p_L + ρ_L⋅u_L            ⎥\n",
+						"⎢                                    ⎥\n", "⎢    ⎛          ⎛                 2⎞⎞⎥\n",
+						"⎢    ⎜          ⎜    p_L       u_L ⎟⎟⎥\n", "⎢u_L⋅⎜p_L + ρ_L⋅⎜─────────── + ────⎟⎟⎥\n",
+						"⎣    ⎝          ⎝ρ_L⋅(γ - 1)    2  ⎠⎠⎦"
+					]
+				},
+				"execution_count" : 3,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"rho_L = Symbol('rho_L')\n", "M = Symbol('M')\n", "u_L = Symbol('u_L')\n", "p_L = Symbol('p_L')\n",
+				"e_L = p_L / (rho_L * (gamma - 1))\n", "E_L = rho_L * (e_L + u_L**2 / 2)\n", "U_L = Matrix([rho_L, rho_L*u_L, E_L])\n", "\n",
+				"F_L = Matrix([rho_L*u_L,\n", "        rho_L*u_L**2 + p_L,\n", "        (E_L + p_L)*u_L])\n", "F_L"
+			]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : ["Now, we want to obtain the left (post-shock) state $U_L$ in terms of the pre-shock state $U_R$ and the Mach number "
+				    "$\\mathcal{M}$, which is defined as $\\mathcal{M} = u_s / c_R$."]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 4,
+			"metadata" : {},
+			"outputs" : [],
+			"source" : [ "M = Symbol('M')\n", "c_R = sqrt(gamma * p_R / rho_R)\n", "u_s = M * c_R" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 5,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "at39zvHCZ3JAtZ6ygnrH3o2W2uPTrGYuuRY3QQA3pjVDWQNHgglTXY6C2TiBWQT2/Uvm+nbkh85deSzyogf8DvDWTuyWQLlEAAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle \\rho_{L} = \\frac{M \\rho_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}{M "
+							"\\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - u_{L}}$"],
+					"text/plain" : [
+						"                 _______ \n", "                ╱ γ⋅p_R  \n", "       M⋅ρ_R⋅  ╱  ─────  \n",
+						"             ╲╱    ρ_R   \n", "ρ_L = ───────────────────\n", "            _______      \n",
+						"           ╱ γ⋅p_R       \n", "      M⋅  ╱  ─────  - u_L\n", "        ╲╱    ρ_R        "
+					]
+				},
+				"execution_count" : 5,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "rho_L_sol1 = solveset(Eq(u_s * (U_L[0] - U_R[0]), F_L[0] - F_R[0]), rho_L).args[0]\n", "Eq(rho_L, rho_L_sol1)" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 6,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "7gq6kDmHpmdryaw0knHzAnmnjKKn3zIYU5C1t62ZOFGNLB3zEmuzrWCqqhn6HF1BJYLq9N1kH5SbDwedvKFVekK83GAgZJFtIMur6FXFdU4MXHoR01"
+					    "cnLmtKu4vq74nu70Ts2yz1Yv3WFOUNHx5pIvYB/f3s3wGpwuv+nwiDnghYSXaLOtELdVqx+TAf4iZNXMdmpz8AAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle \\rho_{L} = \\frac{p_{L} - p_{R}}{u_{L} \\left(M \\sqrt{\\frac{\\gamma "
+							"p_{R}}{\\rho_{R}}} - u_{L}\\right)}$"],
+					"text/plain" : [
+						"              p_L - p_R        \n", "ρ_L = ─────────────────────────\n", "          ⎛      _______      ⎞\n",
+						"          ⎜     ╱ γ⋅p_R       ⎟\n", "      u_L⋅⎜M⋅  ╱  ─────  - u_L⎟\n", "          ⎝  ╲╱    ρ_R        ⎠"
+					]
+				},
+				"execution_count" : 6,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "rho_L_sol2 = solveset(Eq(u_s * (U_L[1] - U_R[1]), F_L[1] - F_R[1]), rho_L).args[0]\n", "Eq(rho_L, rho_L_sol2)" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 7,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "Yby95OgZ74u67QfKcLRHLwlyCZ6tkacQS+riIsFxQDAA++Br3ZSjU81M6vNDKqvpx0zHQ7xW2fT7y4BbF0nql+"
+					    "JZlReIewHmzBy7fTh81HrGS8GKMLTI7nXr08eNVvlTfacjeIsSs/"
+					    "GsNN0ECtZt9NJQD7w7xDtFzxZ2kCFQei4t9OhSLw0wKOG9mEs6LLUqBOJ+Rb7jfqz0prpQWZcd0ZGIWriYYxz+"
+					    "rVjNuek3L5lRbUUgoJvbacEG6IlmWxLOxOW9mk84LLkQBjUGTa3YQqRqzu++qSbZMATEP6wKB/nNk5DKIz4/"
+					    "F9i4wz1FODuuhgPNmPbzYa0uQCezvVrvtjGCulIwSV7iLcVhIvIuD++omICpTO37NXo7DSijgvFkJI3baDI0/rCQUkn1AehYl/"
+					    "PTdLLItV0gMvBnXXdLr8F5SFIJJtAdXQAHnzQqYsMcmRHlg1vpsErhSmk26XRdkJXTy4Ns1Bc/"
+					    "XeefN+WjrmC9Agf8H69oPcfYH68sAAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle \\rho_{L} = \\frac{2 \\left(- M p_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + M "
+							"p_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + \\gamma p_{L} u_{L}\\right)}{u_{L}^{2} \\left(M "
+							"\\gamma \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - "
+							"\\gamma u_{L} + u_{L}\\right)}$"],
+					"text/plain" : [
+						"        ⎛            _______             _______            ⎞\n",
+						"        ⎜           ╱ γ⋅p_R             ╱ γ⋅p_R             ⎟\n",
+						"      2⋅⎜- M⋅p_L⋅  ╱  ─────  + M⋅p_R⋅  ╱  ─────  + γ⋅p_L⋅u_L⎟\n",
+						"        ⎝        ╲╱    ρ_R           ╲╱    ρ_R              ⎠\n",
+						"ρ_L = ───────────────────────────────────────────────────────\n",
+						"             ⎛        _______         _______              ⎞ \n",
+						"           2 ⎜       ╱ γ⋅p_R         ╱ γ⋅p_R               ⎟ \n",
+						"        u_L ⋅⎜M⋅γ⋅  ╱  ─────  - M⋅  ╱  ─────  - γ⋅u_L + u_L⎟ \n",
+						"             ⎝    ╲╱    ρ_R       ╲╱    ρ_R                ⎠ "
+					]
+				},
+				"execution_count" : 7,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "rho_L_sol3 = solveset(Eq(u_s * (U_L[2] - U_R[2]), F_L[2] - F_R[2]), rho_L).args[0]\n", "Eq(rho_L, rho_L_sol3)" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 8,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "vDkdPUSD5n4jGn86ccRdPrukuhCd3S47XIAkcx0u8O5r/"
+					    "ySXfMEUon8qlhy8Ii4+"
+					    "gkmf0a2zK4xQRcCH0oZAjwAvd5YeKMUchjqASIZByHEchnH0PdKQ87qkj8LDOdu7WEIiaDUErv4uDxoPsGeK70N1v9bHFPIH7DyPgQngYo62kQJvx6"
+					    "RA2Y3LioYzyjy4tHgFd4u/WrLxNXt0c3eRtr3YagUprPlJIIDE3ca1/"
+					    "fopxJGvkgeE0DgHXhOPwusjUEijM0Cdyj6yDkcdX9ngssWaKIrT+GpsBNuPqmnAGeBeUFYGCOA/"
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+					"text/latex" : ["$\\displaystyle u_{L} = \\frac{M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} \\left(\\rho_{L} - "
+							"\\rho_{R}\\right)}{\\rho_{L}}$"],
+					"text/plain" : [
+						"            _______            \n", "           ╱ γ⋅p_R             \n", "      M⋅  ╱  ───── ⋅(ρ_L - ρ_R)\n",
+						"        ╲╱    ρ_R              \n", "u_L = ─────────────────────────\n", "                 ρ_L           "
+					]
+				},
+				"execution_count" : 8,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "u_L_sol1 = solveset(Eq(u_s * (U_L[0] - U_R[0]), F_L[0] - F_R[0]), u_L).args[0]\n", "Eq(u_L, u_L_sol1)" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 9,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
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+					"text/latex" : ["$\\displaystyle u_{L} = \\frac{M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}{2} - "
+							"\\frac{\\sqrt{\\frac{\\rho_{L} \\left(M^{2} \\gamma p_{R} \\rho_{L} - 4 \\rho_{R} \\left(p_{L} - "
+							"p_{R}\\right)\\right)}{\\rho_{R}}}}{2 \\rho_{L}}$"],
+					"text/plain" : [
+						"                           ________________________________________\n",
+						"            _______       ╱     ⎛ 2                              ⎞ \n",
+						"           ╱ γ⋅p_R       ╱  ρ_L⋅⎝M ⋅γ⋅p_R⋅ρ_L - 4⋅ρ_R⋅(p_L - p_R)⎠ \n",
+						"      M⋅  ╱  ─────      ╱   ────────────────────────────────────── \n",
+						"        ╲╱    ρ_R     ╲╱                     ρ_R                   \n",
+						"u_L = ───────────── - ─────────────────────────────────────────────\n",
+						"            2                             2⋅ρ_L                    "
+					]
+				},
+				"execution_count" : 9,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "u_L_sol2 = solveset(Eq(u_s * (U_L[1] - U_R[1]), F_L[1] - F_R[1]), u_L).args[0]\n", "Eq(u_L, u_L_sol2)" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 10,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "ERG7lFfN1A6fsi7hC81hIb4n2Uh4AS2rP6YozZoYdciCbZuEM6UP+BGs8P+BUmO+"
+					    "XpHaeeCWKfe2ipLl7Uj4FvItffg6fVPv0yxE9lMSRQQom1TD9Y0kijbzjIN5MoJJPXjkIFnF0dg5wTmg8DIgjfk+T3cJCKSMfuXvfHfm6/"
+					    "is8W0d7z4ITnpTdAK2UoSZ/"
+					    "JfCVghfl8fAv5byPX12eQ1FjHYD2L5suwkP+xXPmhf2NM28fWOyUH3DCdBwDWwSWBcfSZ2Gmja2BQNWtTXDaZokOexPATciL+"
+					    "8PpmjRhDXpB8wlAaGLW1KQpwDFy9z4Qg4gS28g85UPexUqc3pTMV6MY7AaQj8H+dTe9VIFrE9AAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle p_{L} = M \\rho_{L} u_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + p_{R} - "
+							"\\rho_{L} u_{L}^{2}$"],
+					"text/plain" : [
+						"                    _______                 \n", "                   ╱ γ⋅p_R                 2\n",
+						"p_L = M⋅ρ_L⋅u_L⋅  ╱  ─────  + p_R - ρ_L⋅u_L \n", "                ╲╱    ρ_R                   "
+					]
+				},
+				"execution_count" : 10,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "p_L_sol1 = solveset(Eq(u_s * (U_L[1] - U_R[1]), F_L[1] - F_R[1]), p_L).args[0]\n", "Eq(p_L, p_L_sol1)\n" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 11,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "Zruap6snL1Fvvjy3pr/7C2whprLY9bEl362oEtozAk48fP25Zf+tuBIyAEZiEQE1AeMo7PI+T5LmwETACRmDPCPw/"
+					    "xzFq4ird+u8AAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle p_{L} = \\frac{- M \\gamma \\rho_{L} u_{L}^{2} \\sqrt{\\frac{\\gamma "
+							"p_{R}}{\\rho_{R}}} + 2 M p_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + M \\rho_{L} u_{L}^{2} "
+							"\\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} + \\gamma \\rho_{L} u_{L}^{3} - \\rho_{L} u_{L}^{3}}{2 "
+							"\\left(M \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - \\gamma u_{L}\\right)}$"],
+					"text/plain" : [
+						"                         _______               _______                  ______\n",
+						"                   2    ╱ γ⋅p_R               ╱ γ⋅p_R             2    ╱ γ⋅p_R\n",
+						"      - M⋅γ⋅ρ_L⋅u_L ⋅  ╱  ─────  + 2⋅M⋅p_R⋅  ╱  ─────  + M⋅ρ_L⋅u_L ⋅  ╱  ─────\n",
+						"                     ╲╱    ρ_R             ╲╱    ρ_R                ╲╱    ρ_R \n",
+						"p_L = ────────────────────────────────────────────────────────────────────────\n",
+						"                                            ⎛      _______        ⎞           \n",
+						"                                            ⎜     ╱ γ⋅p_R         ⎟           \n",
+						"                                          2⋅⎜M⋅  ╱  ─────  - γ⋅u_L⎟           \n",
+						"                                            ⎝  ╲╱    ρ_R          ⎠           \n", "\n",
+						"_                        \n", "             3          3\n", "  + γ⋅ρ_L⋅u_L  - ρ_L⋅u_L \n",
+						"                         \n", "─────────────────────────\n", "                         \n",
+						"                         \n", "                         \n", "                         "
+					]
+				},
+				"execution_count" : 11,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [ "p_L_sol2 = solveset(Eq(u_s * (U_L[2] - U_R[2]), F_L[2] - F_R[2]), p_L).args[0]\n", "Eq(p_L, p_L_sol2)\n" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 12,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "hHqFfapDiAAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle u_{L} = \\frac{p_{L} - p_{R}}{M \\rho_{R} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}$"],
+					"text/plain" : [
+						"          p_L - p_R    \n", "u_L = ─────────────────\n", "                _______\n",
+						"               ╱ γ⋅p_R \n", "      M⋅ρ_R⋅  ╱  ───── \n", "            ╲╱    ρ_R  "
+					]
+				},
+				"execution_count" : 12,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# write u_L in terms of U_R and p_L\n", "u_L_sol_new1 = solveset(Eq(rho_L_sol1, rho_L_sol2), u_L).args[0].args[0]\n",
+				"Eq(u_L, u_L_sol_new1)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 13,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "+BtSzgRJYInGTZa3c6KwSSJwlBvm90HA+TEK9j3J5tPxf7xrBqI52+PgRMnteHvWs+HAJ4j/"
+					    "x6jk2iauBfEnmLWNuGEcS68y+n1cJO542Ap+152z+X3kOE5ZomnRZhMi3nuM95NaRriGplqnmcNgL2PD0GjhoBESHT9bs6bkdHk4z/6uL/"
+					    "6Num7JCt3xgWgDluRcCeZyssFh4RAhAhofq2MHbcH3cQJ3lb1zsT6fqNYSDksDF5ehAcOwLxtjDWNd/"
+					    "07CyE6zeG9QQr12yetudq+Xz6DRH2XruUd8k6KKFWRnK/"
+					    "MWyLiz8TAvY8PRSOFoE0zWa988q3hQGC8vuNYUc7GqbvmMlzekytcTkIPEhNqXmRXc0TeTKt7zu171JjeSYIeNqeiaEz7SY/"
+					    "v3woUmw+35kpHO72lAj8D+yq72MYA3oTAAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle p_{L} = \\frac{M^{2} \\gamma p_{R} \\rho_{L} - M^{2} \\gamma p_{R} \\rho_{R} + p_{R} "
+							"\\rho_{L}}{\\rho_{L}}$"],
+					"text/plain" : [
+						"       2              2                    \n", "      M ⋅γ⋅p_R⋅ρ_L - M ⋅γ⋅p_R⋅ρ_R + p_R⋅ρ_L\n",
+						"p_L = ─────────────────────────────────────\n", "                       ρ_L                 "
+					]
+				},
+				"execution_count" : 13,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# write p_L in terms of U_R and rho_L\n", "p_L_sol_new1 = solveset(Eq(u_L_sol1, u_L_sol2), p_L).args[2].args[0]\n",
+				"Eq(p_L, p_L_sol_new1)\n"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 14,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "sjtDPVzlttAwHy4jX52K2dCwILaTMA6WyNwaQSkyeAMWPpuGmfDTEbg7AiYD88OuQtcGQK+"
+					    "9bmyDnVzjECBQPhuWlwsiyA7jcDZEDAfng1qF7Q2BCyora1H3R4j0EaA/4NNW0/tELuMwPkQMB+eD2uXtDIE/"
+					    "g92fUQ5lvt0bgAAAABJRU5ErkJggg==",
+					"text/latex" : ["$\\displaystyle \\rho_{L} = \\frac{2 \\gamma p_{R} \\rho_{R}}{2 M^{2} \\gamma p_{R} - M \\gamma "
+							"\\rho_{R} u_{L} \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} - 3 M \\rho_{R} u_{L} \\sqrt{\\frac{\\gamma "
+							"p_{R}}{\\rho_{R}}} + \\gamma \\rho_{R} u_{L}^{2} + \\rho_{R} u_{L}^{2}}$"],
+					"text/plain" : [
+						"                                           2⋅γ⋅p_R⋅ρ_R                        \n",
+						"ρ_L = ────────────────────────────────────────────────────────────────────────\n",
+						"                                   _______                   _______          \n",
+						"         2                        ╱ γ⋅p_R                   ╱ γ⋅p_R           \n",
+						"      2⋅M ⋅γ⋅p_R - M⋅γ⋅ρ_R⋅u_L⋅  ╱  ─────  - 3⋅M⋅ρ_R⋅u_L⋅  ╱  ─────  + γ⋅ρ_R⋅u\n",
+						"                               ╲╱    ρ_R                 ╲╱    ρ_R            \n", "\n", "              \n",
+						"──────────────\n", "              \n", "  2          2\n", "_L  + ρ_R⋅u_L \n", "              "
+					]
+				},
+				"execution_count" : 14,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# write rho_L in terms of U_R and u_L\n", "rho_L_sol_new1 = solveset(Eq(p_L_sol1, p_L_sol2), rho_L).args[0]\n",
+				"Eq(rho_L, rho_L_sol_new1)"
+			]
+		},
+		{"cell_type" : "markdown", "metadata" : {}, "source" : ["We finally obtain:"]},
+		{
+			"cell_type" : "code",
+			"execution_count" : 15,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "wBdub5MFoJHTQAAAABJRU5ErkJggg==",
+					"text/latex" : ["$\\displaystyle p_{L} = \\frac{p_{R} \\left(2 M^{2} \\gamma - \\gamma + 1\\right)}{\\gamma + 1}$"],
+					"text/plain" : [
+						"          ⎛   2          ⎞\n", "      p_R⋅⎝2⋅M ⋅γ - γ + 1⎠\n", "p_L = ────────────────────\n",
+						"             γ + 1        "
+					]
+				},
+				"execution_count" : 15,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# substitute everything\n",
+				"p_L_sol_final = simplify(solveset(Eq(p_L, p_L_sol_new1.subs(rho_L, rho_L_sol_new1).subs(u_L, u_L_sol_new1)), p_L).args[0])\n",
+				"p_L_lambda = lambdify([M, p_R, gamma], p_L_sol_final)\n", "Eq(p_L, p_L_sol_final)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 16,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
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+					    "Vwqf5sC9C87q2h1i9Zq3z0EvdMoasLrvut/"
+					    "4LByEfQAfoDLGRUFVpEVNx+7ESDG4o8KFvOqJjZPBIfuwFlr1xc+y2ZsYrfKpMvyYGNvsfQ2luSrSUI4dR30uWft+"
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+					"text/latex" : ["$\\displaystyle u_{L} = \\frac{2 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}} \\left(M^{2} - 1\\right)}{M "
+							"\\left(\\gamma + 1\\right)}$"],
+					"text/plain" : [
+						"            _______         \n", "           ╱ γ⋅p_R  ⎛ 2    ⎞\n", "      2⋅  ╱  ───── ⋅⎝M  - 1⎠\n",
+						"        ╲╱    ρ_R           \n", "u_L = ──────────────────────\n", "            M⋅(γ + 1)       "
+					]
+				},
+				"execution_count" : 16,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# back substitute\n", "u_L_sol_final = simplify(u_L_sol_new1.subs(p_L, p_L_sol_final))\n",
+				"u_L_lambda = lambdify([M, rho_R, p_R, gamma], u_L_sol_final)\n", "Eq(u_L, u_L_sol_final)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 17,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					"text/latex" :
+					    ["$\\displaystyle \\rho_{L} = \\frac{M^{2} \\rho_{R} \\left(\\gamma + 1\\right)}{M^{2} \\gamma - M^{2} + 2}$"],
+					"text/plain" : [
+						"       2            \n", "      M ⋅ρ_R⋅(γ + 1)\n", "ρ_L = ──────────────\n", "       2      2     \n",
+						"      M ⋅γ - M  + 2 "
+					]
+				},
+				"execution_count" : 17,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# back substitute\n", "rho_L_sol_final = simplify(rho_L_sol_new1.subs(u_L, u_L_sol_final))\n",
+				"rho_L_lambda = lambdify([M, rho_R, gamma], rho_L_sol_final)\n", "Eq(rho_L, rho_L_sol_final)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 65,
+			"metadata" : {},
+			"outputs" : [ {
+				"name" : "stdout",
+				"output_type" : "stream",
+				"text" : [
+					"const Real x0 = std::pow(M, 2);\n", "const Real x1 = gamma + 1;\n", "const Real x2 = gamma*x0;\n",
+					"const Real x3 = 1.0/x1;\n", "const Real x4 = std::sqrt(gamma*p_R/rho_R);\n",
+					"const Real rho_L = rho_R*x0*x1/(-x0 + x2 + 2);\n", "const Real u_L = 2*x3*x4*(x0 - 1)/M;\n",
+					"const Real P_L = p_R*x3*(-gamma + 2*x2 + 1);\n", "const Real v_shock = M*x4;\n"
+				]
+			} ],
+			"source" : [
+				"## cxxcode\n", "vars = ['rho_L', 'u_L', 'P_L', 'v_shock']\n", "exprs = [rho_L_sol_final, u_L_sol_final, p_L_sol_final, u_s]\n",
+				"\n", "optim_exprs = cse(exprs)\n", "expr1 = optim_exprs[0]\n", "expr2 = optim_exprs[1]\n", "for var, expr in expr1:\n",
+				"    print(\"const Real \" + cxxcode(expr, assign_to=var))\n", "for var, expr in zip(vars, expr2):\n",
+				"    print(\"const Real \" + cxxcode(expr, assign_to=var))"
+			]
+		},
+		{
+			"cell_type" : "markdown",
+			"metadata" : {},
+			"source" : [
+				"### Converting between parameterizations\n", "\n",
+				"We want convert between (pre-shock, right-side) background density $\\rho_R$, background pressure $p_R$ and wind velocity "
+				"$v_{\\text{wind}}$ and Mach number $\\mathcal{M}$.\n",
+				"\n", "That is:\n", "\n", "$\\rho_{bg} = \\rho_{R}$,\n", "\n", "$p_{bg} = p_R$,\n", "\n", "and\n", "\n",
+				"$v_{\\text{wind}} = u_L$."
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 18,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
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+					    "Frngz7KEeoTh0w7g+urjskIGAEjYASMgBEwAkbACCyOwP8Dl+k3qlycesYAAAAASUVORK5CYII=",
+					"text/latex" :
+					    ["$\\displaystyle \\left\\{\\frac{u_{L} \\left(\\gamma + 1\\right)}{4 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}} - "
+					     "\\frac{\\sqrt{\\frac{16 \\gamma p_{R}}{\\rho_{R}} + u_{L}^{2} \\left(\\gamma + 1\\right)^{2}}}{4 "
+					     "\\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}, \\frac{u_{L} \\left(\\gamma + 1\\right)}{4 \\sqrt{\\frac{\\gamma "
+					     "p_{R}}{\\rho_{R}}}} + \\frac{\\sqrt{\\frac{16 \\gamma p_{R}}{\\rho_{R}} + u_{L}^{2} \\left(\\gamma + "
+					     "1\\right)^{2}}}{4 \\sqrt{\\frac{\\gamma p_{R}}{\\rho_{R}}}}\\right\\} \\setminus \\left\\{0\\right\\}$"],
+					"text/plain" : [
+						"⎧                    __________________________                      _________\n",
+						"⎪                   ╱ 16⋅γ⋅p_R      2        2                      ╱ 16⋅γ⋅p_R\n",
+						"⎪                  ╱  ──────── + u_L ⋅(γ + 1)                      ╱  ────────\n",
+						"⎪ u_L⋅(γ + 1)    ╲╱     ρ_R                       u_L⋅(γ + 1)    ╲╱     ρ_R   \n",
+						"⎨───────────── - ──────────────────────────────, ───────────── + ─────────────\n",
+						"⎪      _______                 _______                 _______                \n",
+						"⎪     ╱ γ⋅p_R                 ╱ γ⋅p_R                 ╱ γ⋅p_R                 \n",
+						"⎪4⋅  ╱  ─────            4⋅  ╱  ─────            4⋅  ╱  ─────            4⋅  ╱\n",
+						"⎩  ╲╱    ρ_R               ╲╱    ρ_R               ╲╱    ρ_R               ╲╱ \n", "\n",
+						"_________________⎫      \n", "      2        2 ⎪      \n", " + u_L ⋅(γ + 1)  ⎪      \n",
+						"                 ⎪      \n", "─────────────────⎬ \\ {0}\n", " _______         ⎪      \n",
+						"╱ γ⋅p_R          ⎪      \n", "  ─────          ⎪      \n", "   ρ_R           ⎭      "
+					]
+				},
+				"execution_count" : 18,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"v_wind = Symbol(r'v_{\\text{wind}}')\n", "rho_bg = Symbol(r'\\rho_{bg}')\n", "p_bg = Symbol(r'p_{bg}')\n",
+				"Mach_solve = solveset(Eq(u_L, u_L_sol_final), M)\n", "Mach_solve"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 19,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
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+					    "A2mok64w/"
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+					    "+ZLrj8UQoH9Io54JwusBy5lrD0sqBRUj3ehcmuN/"
+					    "Pw5lfWIc6AwB3xuFGbopYJzC2m7nstdEwcpAAR+yf0c3HHWxiBVaje49FQAC+p5q2DAR2XCfpTuJfFrNeWPzoHEARu3eAwOGnc+NxJrriviCmmj/"
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+					    "nw8hbviYOaKz53LimDh+g1feQBhhzomR85Sgk/"
+					    "trnrSZhFcoo0upfh42M8NtZOOBz4yxsr6tRV0jb9of5yrFAcNlVE6Qc8+PhYwcjqsHZEdkVB3xu7Ko7lxHje0jL+"
+					    "LaoVrSI7OTaq0VAvJJzYIcc8Lmxw05dQJJbSAuYtrIKrolDnIArQXl158CuOOBzY1fdOZ8YV0jzeba2BgcFzD2xFpbXdw7siQM+N/"
+					    "bUmwto+T8MAKEGb/7W7gAAAABJRU5ErkJggg==",
+					"text/latex" : ["$\\displaystyle M = \\frac{v_{\\text{wind}} \\left(\\gamma + 1\\right)}{4 \\sqrt{\\frac{\\gamma "
+							"p_{bg}}{\\rho_{bg}}}} + \\frac{\\sqrt{v_{\\text{wind}}^{2} \\left(\\gamma + 1\\right)^{2} + \\frac{16 "
+							"\\gamma p_{bg}}{\\rho_{bg}}}}{4 \\sqrt{\\frac{\\gamma p_{bg}}{\\rho_{bg}}}}$"],
+					"text/plain" : [
+						"                                  _________________________________________\n",
+						"                                 ╱                2        2   16⋅γ⋅p_{bg} \n",
+						"                                ╱  v_{\\text{wind}} ⋅(γ + 1)  + ─────────── \n",
+						"    v_{\\text{wind}}⋅(γ + 1)   ╲╱                                \\rho_{bg}  \n",
+						"M = ─────────────────────── + ─────────────────────────────────────────────\n",
+						"             ___________                          ___________              \n",
+						"            ╱  γ⋅p_{bg}                          ╱  γ⋅p_{bg}               \n",
+						"       4⋅  ╱  ─────────                     4⋅  ╱  ─────────               \n",
+						"         ╲╱   \\rho_{bg}                       ╲╱   \\rho_{bg}               "
+					]
+				},
+				"execution_count" : 19,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# use positive root (only physical solution)\n",
+				"Mach_sol = Mach_solve.args[0].args[1].subs(u_L, v_wind).subs(rho_R, rho_bg).subs(p_R, p_bg)\n",
+				"compute_mach_number = lambdify([v_wind, rho_bg, p_bg, gamma], Mach_sol)\n", "Eq(M, Mach_sol)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 28,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
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+					    "8o+vJbvsKAXSfZvUi7Xc49EtAVxchfK8M/IuxBg35UsRKf8paCWGbfoot4khAzy8KejDfvPrs9mHlYF+j+5b6UaBxKIe6fF9RDOBhjTT7dujzoalq/"
+					    "NojDi5JJkF6OO4naUtCM77G8kNu4VsXr1K0CbqL94Ml699tN3Q69SR8Va008ZTZzvRXpLuEJ6fKTUciIDXaC/ou2/"
+					    "KNMJLOwPqXV+rduI9s34Mm4RbwRLj2Y4itukzjO1T6/QjAOfeGUDJTqPBzU5L902q9bN9stEokKAcReswU0XEe6Wr7//"
+					    "idz6NWGeUAQY+F1ohk3ps6AL+i3jqHCwKSl9jG3g5sMQDBk2+7y9N59oNnvKd2gudX2gF34q1lNc/"
+					    "OF6jnRyQ2uBzvfrU25U3S8yCSbVOXk6St0m63pNJfZR/KKPHMRcbQL3yetB26d5DtxcchtoP7TpYvCvLMB+H0kKnIynYf/"
+					    "X7KJ5vzUDJQagTILtyGyCecrWzrIn++ZXWkoE5ZlA33wCrv6mOwCUbLk03COC/"
+					    "8lhyMqLtc70wCvY1Kss+ssEwZqUudZCpm75Qn2K78nb9yX1wEB5eu7SjfyhOSYlae/9K8SO/htiLKX20gU/"
+					    "nP1MyDW2v4D6Blydcr+"
+					    "6zdFf8728W13AGhhM0ngZLWZXazOSrSjiHfTcx6BXMOT5VZsHpXwggg7oIyqnRlU0ZNWJ5UnhfdgOzQ9LPAN4A0R6uI5hI3oH3b2fOdawFOTiXpgtm"
+					    "rCvvOAlNFJ/tg9+w/0DrdX3yKRma+B7SXsLk33B6eUPXtQ9rdimglCH4q1Pz72Kd+uibDynaJaCpU6/uc3R/A7pKZo4YflDJAAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle M = 1.42225795167899$"],
+					"text/plain" : ["M = 1.42225795167899"]
+				},
+				"execution_count" : 28,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# numerical example\n", "\n", "import numpy as np\n", "mH = 1.673533e-24 # H mass (g)\n",
+				"kB = 1.380649e-16 # Boltzmann constant (cgs)\n", "gamma_bg = 5./3.  # assume gamma is constant\n", "\n",
+				"v_wind_kms = 200. # km/s\n", "nH_bg = 1.0e-3 # cm^-3\n", "rho_bg = nH_bg * mH # mu = 1 for OOM estimate\n",
+				"P_bg = 1.0e4 # K cm^-3\n", "\n", "# compute Mach number of shock\n",
+				"Mach_shock = compute_mach_number(v_wind_kms * 1.0e5, nH_bg * mH, P_bg * kB, 5./3.)\n", "Eq(M, Mach_shock)"
+			]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 24,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" : "iVBORw0KGgoAAAANSUhEUgAAAGIAAAARCAYAAAAmE3lhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADZUlEQVRYCe2Y61EUQRCAD4oAKMgA"
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+						      "BjGwvV/wLjpxU4BC8U8vyGEckY0gvZRq1eGJS41r5JBMYfMfCN4gD6nNElApkJOgA/"
+						      "tjkeLwbM6TlstcwDBnFuX7AxCR5J8Sa10S46rbhMVq1eO90EqrJfax30TBR8yyhbVdnQOTrC1iB6KZpAL7grZJ/"
+						      "AVYBu70KtMlpcya6+mMc/9Dcrl6vVe2y6TTvCszQC/Qb+pNA22KWsUcHukmGlebHNXaXFOk/"
+						      "G4pud5z6zj2XB5fWjiv1fUauX7TNdZb+Kc01rgg24rZiPJVtZWffeDV2CHTPtean/V3n25dHqWd/"
+						      "9p9AB+GbfkO6vhJDlCz9kWTdkcbnnZw1du44dEWC15IWVe+"
+						      "TodHSMbAkmcKEjhnmf9I5gfv3fBnedAG1xjcAGzwJqeGUJoiOiwGr10hQ9cqZ9ToQOlQHvqh3Hrayv4PtYAtpjSQiHH7jKX+"
+						      "wXSmDN9K1vu+C4nMPM5ETXe2/E5414Ln7J8KUg9lqrl+fI9Ez71aTda692Izs61OpYWeFYmNkN98h7tvHwX2H8sajcvH/e/"
+						      "NTRDWRH8M0+wCbkDtx0PvQIeh3kXXkoLyCr1XO9McPPIR3U2OeOsHLOMbJCvKQMrpXhBzC/0XxhlNB1TH6AvpvZAS/"
+						      "UKXmuBenv2JkMu7iEKKyQxx53Ebhv8St8n1sPmyjKHzF5wlPX+"
+						      "aOPfizsp1srLFq9WRfeZPohreyg5NczmSOQj6Ysn0kT5MH7AbltvvWchJkh7Cnko6n3YBpDkD2q3rY6doRkcxyBTVDvVdGHzzA9Ar8BR"
+						      "WCJ/3OY4W0AAAAASUVORK5CYII=",
+					"text/latex" : ["$\\displaystyle u_{L} = 200.0$"],
+					"text/plain" : ["u_L = 200.0"]
+				},
+				"execution_count" : 24,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" :
+			    [ "# compute post-shock fluid velocity in km/s\n", "Eq(u_L, u_L_lambda(Mach_shock, nH_bg * mH, P_bg * kB, 5./3.) / 1.0e5)" ]
+		},
+		{
+			"cell_type" : "code",
+			"execution_count" : 26,
+			"metadata" : {},
+			"outputs" : [ {
+				"data" : {
+					"image/png" :
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+					    "JOGxyOipl+47ZYvPFuLoUv8bOBvETtg77mluTR+N+4NHQkUwb+lYFf2qisirqDwbnnA3J9wNka+g4VXSY1ccX82JE+"
+					    "l43pRsDK6kVtUDmaWLEJ90yV3cyX+3TOV0ZewKWBKavbH/N0Km+jLWlJ7g6B5T/xQQiWugtFvKKq7CJ8AF8+pJI2yS34qeP9aOg7Z7Co/"
+					    "haJGPnpdOexscRz4AOnLRxqqmH6+RlbFgQ+qe89JP7YW6Vz5AhP20z6zdommyeeU3GVUkgBbdCZmpFSkch66y0Zuvop+BWvnu6vQV+"
+					    "NQsc7KiwCZPumSSNlSVdBXbktV+2t8Ava4FdTyDv5143PO4FT6PFDwhb0v3fvQUujgX+AezqiHJAZpORAAAAAElFTkSuQmCC",
+					"text/latex" : ["$\\displaystyle u_{s} = 527.384360142785$"],
+					"text/plain" : ["uₛ = 527.384360142785"]
+				},
+				"execution_count" : 26,
+				"metadata" : {},
+				"output_type" : "execute_result"
+			} ],
+			"source" : [
+				"# compute shock velocity in km/s\n", "u_s_lambda = lambdify([M, rho_R, p_R, gamma], u_s)\n",
+				"Eq(Symbol('u_s'), u_s_lambda(Mach_shock, nH_bg * mH, P_bg * kB, 5./3.) / 1.0e5)"
+			]
+		}
+	],
+	"metadata" : {
+		"kernelspec" : {"display_name" : "Python 3", "language" : "python", "name" : "python3"},
+		"language_info" : {
+			"codemirror_mode" : {"name" : "ipython", "version" : 3},
+			"file_extension" : ".py",
+			"mimetype" : "text/x-python",
+			"name" : "python",
+			"nbconvert_exporter" : "python",
+			"pygments_lexer" : "ipython3",
+			"version" : "3.11.5"
+		},
+		"orig_nbformat" : 4
+	},
+	"nbformat" : 4,
+	"nbformat_minor" : 2
 }
diff --git a/src/radiation/radiation_system.hpp b/src/radiation/radiation_system.hpp
index 5e9bf9dea..ba439257e 100644
--- a/src/radiation/radiation_system.hpp
+++ b/src/radiation/radiation_system.hpp
@@ -262,15 +262,15 @@ template <typename problem_t> class RadSystem : public HyperbolicSystem<problem_
 	AMREX_GPU_HOST_DEVICE static auto ComputeEnergyMeanOpacity(double rho, double Tgas) -> Real;
 
 	// For multi-group RHD, use DefineOpacityExponentsAndLowerValues to define the opacities.
-	AMREX_GPU_HOST_DEVICE static auto DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> rad_boundaries, double rho,
-									       double Tgas) -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>;
+	AMREX_GPU_HOST_DEVICE static auto DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> rad_boundaries, double rho, double Tgas)
+	    -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>;
 
 	AMREX_GPU_HOST_DEVICE static auto ComputeGroupMeanOpacity(amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> const &kappa_expo_and_lower_value,
 								  amrex::GpuArray<double, nGroups_> const &radBoundaryRatios,
 								  amrex::GpuArray<double, nGroups_> const &alpha_quant) -> quokka::valarray<double, nGroups_>;
-	AMREX_GPU_HOST_DEVICE static auto
-	ComputeBinCenterOpacity(amrex::GpuArray<double, nGroups_ + 1> rad_boundaries,
-				amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> kappa_expo_and_lower_value) -> quokka::valarray<double, nGroups_>;
+	AMREX_GPU_HOST_DEVICE static auto ComputeBinCenterOpacity(amrex::GpuArray<double, nGroups_ + 1> rad_boundaries,
+								  amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> kappa_expo_and_lower_value)
+	    -> quokka::valarray<double, nGroups_>;
 	// AMREX_GPU_HOST_DEVICE static auto
 	// ComputeGroupMeanOpacityWithMinusOneSlope(amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> kappa_expo_and_lower_value,
 	// 					 amrex::GpuArray<double, nGroups_> radBoundaryRatios) -> quokka::valarray<double, nGroups_>;
@@ -280,22 +280,22 @@ template <typename problem_t> class RadSystem : public HyperbolicSystem<problem_
 	AMREX_GPU_HOST_DEVICE static auto
 	ComputeDiffusionFluxMeanOpacity(quokka::valarray<double, nGroups_> kappaPVec, quokka::valarray<double, nGroups_> kappaEVec,
 					quokka::valarray<double, nGroups_> fourPiBoverC, amrex::GpuArray<double, nGroups_> delta_nu_kappa_B_at_edge,
-					amrex::GpuArray<double, nGroups_> delta_nu_B_at_edge,
-					amrex::GpuArray<double, nGroups_ + 1> kappa_slope) -> quokka::valarray<double, nGroups_>;
-	AMREX_GPU_HOST_DEVICE static auto ComputeFluxInDiffusionLimit(amrex::GpuArray<double, nGroups_ + 1> rad_boundaries, double T,
-								      double vel) -> amrex::GpuArray<double, nGroups_>;
+					amrex::GpuArray<double, nGroups_> delta_nu_B_at_edge, amrex::GpuArray<double, nGroups_ + 1> kappa_slope)
+	    -> quokka::valarray<double, nGroups_>;
+	AMREX_GPU_HOST_DEVICE static auto ComputeFluxInDiffusionLimit(amrex::GpuArray<double, nGroups_ + 1> rad_boundaries, double T, double vel)
+	    -> amrex::GpuArray<double, nGroups_>;
 
 	template <typename ArrayType>
-	AMREX_GPU_HOST_DEVICE static auto
-	ComputeRadQuantityExponents(ArrayType const &quant, amrex::GpuArray<double, nGroups_ + 1> const &boundaries) -> amrex::GpuArray<double, nGroups_>;
+	AMREX_GPU_HOST_DEVICE static auto ComputeRadQuantityExponents(ArrayType const &quant, amrex::GpuArray<double, nGroups_ + 1> const &boundaries)
+	    -> amrex::GpuArray<double, nGroups_>;
 
 	AMREX_GPU_HOST_DEVICE static void SolveLinearEqs(JacobianResult<problem_t> const &jacobian, double &x0, quokka::valarray<double, nGroups_> &xi);
 
 	AMREX_GPU_HOST_DEVICE static auto Solve3x3matrix(double C00, double C01, double C02, double C10, double C11, double C12, double C20, double C21,
 							 double C22, double Y0, double Y1, double Y2) -> std::tuple<amrex::Real, amrex::Real, amrex::Real>;
 
-	AMREX_GPU_HOST_DEVICE static auto ComputePlanckEnergyFractions(amrex::GpuArray<double, nGroups_ + 1> const &boundaries,
-								       amrex::Real temperature) -> quokka::valarray<amrex::Real, nGroups_>;
+	AMREX_GPU_HOST_DEVICE static auto ComputePlanckEnergyFractions(amrex::GpuArray<double, nGroups_ + 1> const &boundaries, amrex::Real temperature)
+	    -> quokka::valarray<amrex::Real, nGroups_>;
 
 	AMREX_GPU_HOST_DEVICE static auto ComputeThermalRadiationSingleGroup(amrex::Real temperature) -> double;
 
@@ -304,9 +304,9 @@ template <typename problem_t> class RadSystem : public HyperbolicSystem<problem_
 
 	AMREX_GPU_HOST_DEVICE static auto ComputeThermalRadiationTempDerivativeSingleGroup(amrex::Real temperature) -> Real;
 
-	AMREX_GPU_HOST_DEVICE static auto
-	ComputeThermalRadiationTempDerivativeMultiGroup(amrex::Real temperature,
-							amrex::GpuArray<double, nGroups_ + 1> const &boundaries) -> quokka::valarray<amrex::Real, nGroups_>;
+	AMREX_GPU_HOST_DEVICE static auto ComputeThermalRadiationTempDerivativeMultiGroup(amrex::Real temperature,
+											  amrex::GpuArray<double, nGroups_ + 1> const &boundaries)
+	    -> quokka::valarray<amrex::Real, nGroups_>;
 
 	AMREX_GPU_DEVICE static auto
 	ComputeDustTemperatureBateKeto(double T_gas, double T_d_init, double rho, quokka::valarray<double, nGroups_> const &Erad, double N_d, double dt,
@@ -347,17 +347,18 @@ template <typename problem_t> class RadSystem : public HyperbolicSystem<problem_
 					   int *p_iteration_failure_counter) -> NewtonIterationResult<problem_t>;
 
 	template <FluxDir DIR>
-	AMREX_GPU_DEVICE static auto
-	ComputeCellOpticalDepth(const quokka::Array4View<const amrex::Real, DIR> &consVar, amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> dx, int i, int j, int k,
-				const amrex::GpuArray<double, nGroups_ + 1> &group_boundaries) -> quokka::valarray<double, nGroups_>;
+	AMREX_GPU_DEVICE static auto ComputeCellOpticalDepth(const quokka::Array4View<const amrex::Real, DIR> &consVar,
+							     amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> dx, int i, int j, int k,
+							     const amrex::GpuArray<double, nGroups_ + 1> &group_boundaries)
+	    -> quokka::valarray<double, nGroups_>;
 
 	AMREX_GPU_DEVICE static auto isStateValid(std::array<amrex::Real, nvarHyperbolic_> &cons) -> bool;
 
 	AMREX_GPU_DEVICE static void amendRadState(std::array<amrex::Real, nvarHyperbolic_> &cons);
 
 	template <FluxDir DIR>
-	AMREX_GPU_DEVICE static auto ComputeRadPressure(double erad_L, double Fx_L, double Fy_L, double Fz_L, double fx_L, double fy_L,
-							double fz_L) -> RadPressureResult;
+	AMREX_GPU_DEVICE static auto ComputeRadPressure(double erad_L, double Fx_L, double Fy_L, double Fz_L, double fx_L, double fy_L, double fz_L)
+	    -> RadPressureResult;
 
 	AMREX_GPU_DEVICE static auto ComputeEddingtonTensor(double fx_L, double fy_L, double fz_L) -> std::array<std::array<double, 3>, 3>;
 };
@@ -365,8 +366,8 @@ template <typename problem_t> class RadSystem : public HyperbolicSystem<problem_
 // Compute radiation energy fractions for each photon group from a Planck function, given nGroups, radBoundaries, and temperature
 // This function enforces that the total fraction is 1.0, no matter what are the group boundaries
 template <typename problem_t>
-AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputePlanckEnergyFractions(amrex::GpuArray<double, nGroups_ + 1> const &boundaries,
-									      amrex::Real temperature) -> quokka::valarray<amrex::Real, nGroups_>
+AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputePlanckEnergyFractions(amrex::GpuArray<double, nGroups_ + 1> const &boundaries, amrex::Real temperature)
+    -> quokka::valarray<amrex::Real, nGroups_>
 {
 	quokka::valarray<amrex::Real, nGroups_> radEnergyFractions{};
 	if constexpr (nGroups_ == 1) {
@@ -408,9 +409,9 @@ template <typename problem_t> AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::C
 
 // define ComputeThermalRadiationMultiGroup, returns the thermal radiation power for each photon group. = a_r * T^4 * radEnergyFractions
 template <typename problem_t>
-AMREX_GPU_HOST_DEVICE auto
-RadSystem<problem_t>::ComputeThermalRadiationMultiGroup(amrex::Real temperature,
-							amrex::GpuArray<double, nGroups_ + 1> const &boundaries) -> quokka::valarray<amrex::Real, nGroups_>
+AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputeThermalRadiationMultiGroup(amrex::Real temperature,
+										   amrex::GpuArray<double, nGroups_ + 1> const &boundaries)
+    -> quokka::valarray<amrex::Real, nGroups_>
 {
 	const double power = radiation_constant_ * std::pow(temperature, 4);
 	const auto radEnergyFractions = ComputePlanckEnergyFractions(boundaries, temperature);
@@ -431,8 +432,9 @@ template <typename problem_t> AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::C
 }
 
 template <typename problem_t>
-AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputeThermalRadiationTempDerivativeMultiGroup(
-    amrex::Real temperature, amrex::GpuArray<double, nGroups_ + 1> const &boundaries) -> quokka::valarray<amrex::Real, nGroups_>
+AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputeThermalRadiationTempDerivativeMultiGroup(amrex::Real temperature,
+												 amrex::GpuArray<double, nGroups_ + 1> const &boundaries)
+    -> quokka::valarray<amrex::Real, nGroups_>
 {
 	// by default, d emission/dT = 4 emission / T
 	auto radEnergyFractions = ComputePlanckEnergyFractions(boundaries, temperature);
@@ -698,9 +700,10 @@ AMREX_GPU_DEVICE auto RadSystem<problem_t>::ComputeMassScalars(ArrayType const &
 
 template <typename problem_t>
 template <FluxDir DIR>
-AMREX_GPU_DEVICE auto
-RadSystem<problem_t>::ComputeCellOpticalDepth(const quokka::Array4View<const amrex::Real, DIR> &consVar, amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> dx, int i,
-					      int j, int k, const amrex::GpuArray<double, nGroups_ + 1> &group_boundaries) -> quokka::valarray<double, nGroups_>
+AMREX_GPU_DEVICE auto RadSystem<problem_t>::ComputeCellOpticalDepth(const quokka::Array4View<const amrex::Real, DIR> &consVar,
+								    amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> dx, int i, int j, int k,
+								    const amrex::GpuArray<double, nGroups_ + 1> &group_boundaries)
+    -> quokka::valarray<double, nGroups_>
 {
 	// compute interface-averaged cell optical depth
 
@@ -1049,9 +1052,9 @@ template <typename problem_t> AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::C
 }
 
 template <typename problem_t>
-AMREX_GPU_HOST_DEVICE auto
-RadSystem<problem_t>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> /*rad_boundaries*/, const double /*rho*/,
-							   const double /*Tgas*/) -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
+AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::DefineOpacityExponentsAndLowerValues(amrex::GpuArray<double, nGroups_ + 1> /*rad_boundaries*/,
+										      const double /*rho*/, const double /*Tgas*/)
+    -> amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2>
 {
 	amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> exponents_and_values{};
 	for (int g = 0; g < nGroups_ + 1; ++g) {
@@ -1145,8 +1148,8 @@ AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputeRadQuantityExponents(Arr
 template <typename problem_t>
 AMREX_GPU_HOST_DEVICE auto
 RadSystem<problem_t>::ComputeGroupMeanOpacity(amrex::GpuArray<amrex::GpuArray<double, nGroups_ + 1>, 2> const &kappa_expo_and_lower_value,
-					      amrex::GpuArray<double, nGroups_> const &radBoundaryRatios,
-					      amrex::GpuArray<double, nGroups_> const &alpha_quant) -> quokka::valarray<double, nGroups_>
+					      amrex::GpuArray<double, nGroups_> const &radBoundaryRatios, amrex::GpuArray<double, nGroups_> const &alpha_quant)
+    -> quokka::valarray<double, nGroups_>
 {
 	amrex::GpuArray<double, nGroups_ + 1> const &alpha_kappa = kappa_expo_and_lower_value[0];
 	amrex::GpuArray<double, nGroups_ + 1> const &kappa_lower = kappa_expo_and_lower_value[1];
@@ -1221,10 +1224,13 @@ template <typename problem_t> AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::P
 }
 
 template <typename problem_t>
-AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputeDiffusionFluxMeanOpacity(
-    const quokka::valarray<double, nGroups_> kappaPVec, const quokka::valarray<double, nGroups_> kappaEVec,
-    const quokka::valarray<double, nGroups_> fourPiBoverC, const amrex::GpuArray<double, nGroups_> delta_nu_kappa_B_at_edge,
-    const amrex::GpuArray<double, nGroups_> delta_nu_B_at_edge, const amrex::GpuArray<double, nGroups_ + 1> kappa_slope) -> quokka::valarray<double, nGroups_>
+AMREX_GPU_HOST_DEVICE auto RadSystem<problem_t>::ComputeDiffusionFluxMeanOpacity(const quokka::valarray<double, nGroups_> kappaPVec,
+										 const quokka::valarray<double, nGroups_> kappaEVec,
+										 const quokka::valarray<double, nGroups_> fourPiBoverC,
+										 const amrex::GpuArray<double, nGroups_> delta_nu_kappa_B_at_edge,
+										 const amrex::GpuArray<double, nGroups_> delta_nu_B_at_edge,
+										 const amrex::GpuArray<double, nGroups_ + 1> kappa_slope)
+    -> quokka::valarray<double, nGroups_>
 {
 	quokka::valarray<double, nGroups_> kappaF{};
 	for (int g = 0; g < nGroups_; ++g) {
diff --git a/src/radiation/source_terms_multi_group.hpp b/src/radiation/source_terms_multi_group.hpp
index d7c5dc2e0..d07865fed 100644
--- a/src/radiation/source_terms_multi_group.hpp
+++ b/src/radiation/source_terms_multi_group.hpp
@@ -166,8 +166,8 @@ AMREX_GPU_DEVICE auto RadSystem<problem_t>::SolveMatterRadiationEnergyExchange(
     double const Egas0, quokka::valarray<double, nGroups_> const &Erad0Vec, double const rho, double const T_d0, int const dust_model, double const coeff_n,
     double const lambda_gd_times_dt, double const dt, amrex::GpuArray<Real, nmscalars_> const &massScalars, int const n_outer_iter,
     quokka::valarray<double, nGroups_> const &work, quokka::valarray<double, nGroups_> const &vel_times_F, quokka::valarray<double, nGroups_> const &Src,
-    amrex::GpuArray<double, nGroups_ + 1> const &rad_boundaries, JacobianFunc ComputeJacobian, int *p_iteration_counter,
-    int *p_iteration_failure_counter) -> NewtonIterationResult<problem_t>
+    amrex::GpuArray<double, nGroups_ + 1> const &rad_boundaries, JacobianFunc ComputeJacobian, int *p_iteration_counter, int *p_iteration_failure_counter)
+    -> NewtonIterationResult<problem_t>
 {
 	// 1. Compute energy exchange
 
diff --git a/src/util/CheckNaN.hpp b/src/util/CheckNaN.hpp
index b81122043..107f73693 100644
--- a/src/util/CheckNaN.hpp
+++ b/src/util/CheckNaN.hpp
@@ -23,8 +23,8 @@ AMREX_GPU_HOST_DEVICE auto CheckSymmetryArray(amrex::Array4<const amrex::Real> c
 
 template <typename T>
 AMREX_GPU_HOST_DEVICE auto CheckSymmetryFluxes(amrex::Array4<const amrex::Real> const & /*arr1*/, amrex::Array4<const amrex::Real> const & /*arr2*/,
-					       amrex::Box const & /*indexRange*/, const int /*ncomp*/,
-					       amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> /*dx*/) -> bool
+					       amrex::Box const & /*indexRange*/, const int /*ncomp*/, amrex::GpuArray<amrex::Real, AMREX_SPACEDIM> /*dx*/)
+    -> bool
 {
 	return true; // problem-specific implementation for test problems
 }
diff --git a/src/util/fextract.cpp b/src/util/fextract.cpp
index 5ed4729c9..a36c805fb 100644
--- a/src/util/fextract.cpp
+++ b/src/util/fextract.cpp
@@ -12,8 +12,8 @@
 
 using namespace amrex; // NOLINT
 
-auto fextract(MultiFab &mf, Geometry &geom, const int idir, const Real slice_coord,
-	      const bool center = false) -> std::tuple<Vector<Real>, Vector<Gpu::HostVector<Real>>>
+auto fextract(MultiFab &mf, Geometry &geom, const int idir, const Real slice_coord, const bool center = false)
+    -> std::tuple<Vector<Real>, Vector<Gpu::HostVector<Real>>>
 {
 	AMREX_D_TERM(Real xcoord = slice_coord;, Real ycoord = slice_coord;, Real zcoord = slice_coord;)
 
diff --git a/src/util/fextract.hpp b/src/util/fextract.hpp
index 2267133f9..2343732a8 100644
--- a/src/util/fextract.hpp
+++ b/src/util/fextract.hpp
@@ -9,7 +9,7 @@
 #include <AMReX_PlotFileUtil.H>
 #include <AMReX_Print.H>
 
-auto fextract(amrex::MultiFab &mf, amrex::Geometry &geom, int idir, amrex::Real slice_coord,
-	      bool center = false) -> std::tuple<amrex::Vector<amrex::Real>, amrex::Vector<amrex::Gpu::HostVector<amrex::Real>>>;
+auto fextract(amrex::MultiFab &mf, amrex::Geometry &geom, int idir, amrex::Real slice_coord, bool center = false)
+    -> std::tuple<amrex::Vector<amrex::Real>, amrex::Vector<amrex::Gpu::HostVector<amrex::Real>>>;
 
 #endif // FEXTRACT_HPP_
\ No newline at end of file