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div_ops.cxx
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div_ops.cxx
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/*
Copyright B.Dudson, J.Leddy, University of York, September 2016
email: benjamin.dudson@york.ac.uk
This file is part of Hermes.
Hermes is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
Hermes is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with Hermes. If not, see <http://www.gnu.org/licenses/>.
*/
#include <mpi.h>
#include "div_ops.hxx"
#include <bout/assert.hxx>
#include <bout/mesh.hxx>
#include <derivs.hxx>
#include <globals.hxx>
#include <output.hxx>
#include <utils.hxx>
#include <cmath>
// Div ( n * a * b x Grad(f) / B ), where n is the quantity being advected
const Field3D Div_n_a_bxGrad_f_B(const Field3D &n, const Field3D &a,
const Field3D &f, bool xflux, bool yderiv) {
Field3D result = 0.0;
// X flux
Field3D Vx = a * mesh->J * DDZ(f, DIFF_C2);
if (yderiv)
Vx -= a * mesh->J * (mesh->g_23 / mesh->g_22) * DDY(f, DIFF_C2);
mesh->communicate(Vx);
if (xflux) {
Vx.applyBoundary("neumann");
} else
Vx.applyBoundary("dirichlet");
Field3D ns = n;
Field3D fs = f;
if (mesh->ShiftXderivs) {
Vx = Vx.shiftZ(true); // Shift to orthogonal coordinates for X derivatives
ns = n.shiftZ(true);
fs = f.shiftZ(true);
}
int xs = mesh->xstart;
int xe = mesh->xend;
if (!xflux) {
// Need to use 1st order on boundaries
if (mesh->firstX()) {
int jx = xs;
for (int jy = mesh->ystart; jy <= mesh->yend; jy++)
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
result(jx, jy, jz) += (ns(jx + 1, jy, jz) * Vx(jx + 1, jy, jz) +
ns(jx, jy, jz) * Vx(jx, jy, jz)) /
(2. * mesh->dx(jx, jy) * mesh->J(jx, jy));
/*
output.write("%d, %d, %d: += %e*%e + %e*%e -> %e\n",
jx, jy, jz,
ns(jx+1,jy,jz), Vx(jx+1,jy,jz),
ns(jx,jy,jz), Vx(jx,jy,jz),
result(jx,jy,jz));
*/
}
xs++;
}
if (mesh->lastX()) {
int jx = xe;
for (int jy = mesh->ystart; jy <= mesh->yend; jy++)
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
result(jx, jy, jz) -= (ns(jx, jy, jz) * Vx(jx, jy, jz) +
ns(jx - 1, jy, jz) * Vx(jx - 1, jy, jz)) /
(2. * mesh->dx(jx, jy) * mesh->J(jx, jy));
// if(jy == mesh->ystart)
// output.write("%d: + %e - %e\n", jx,
// n(jx,jy,jz)*Vx(jx,jy,jz), n(jx-1,jy,jz)*Vx(jx-1,jy,jz));
}
xe--;
}
}
// output << "RESULT X1: " << result(2,2,0) << endl;
for (int jx = xs; jx <= xe; jx++) {
for (int jy = mesh->ystart; jy <= mesh->yend; jy++) {
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
int jxm = jx - 1;
if (mesh->firstX() && (jx == mesh->xstart))
jxm = jx;
int jxp = jx + 1;
if (mesh->lastX() && (jx == mesh->xend))
jxp = jx;
result(jx, jy, jz) += (ns(jxp, jy, jz) * Vx(jxp, jy, jz) -
ns(jxm, jy, jz) * Vx(jxm, jy, jz)) /
(2. * mesh->dx(jx, jy) * mesh->J(jx, jy));
// if(jy == mesh->ystart)
// output.write("%d: + %e - %e\n", jx,
// n(jx+1,jy,jz)*Vx(jx+1,jy,jz), n(jx-1,jy,jz)*Vx(jx-1,jy,jz) );
// result(jx, jy, jz) += ( Vx(jx,jy,jz)*(n(jx+1,jy,jz) - n(jx-1,jy,jz))
// + n(jx,jy,jz)*(Vx(jx+1,jy,jz) - Vx(jx-1,jy,jz)) ) / (2.*mesh->dx(jx,
// jy) * mesh->J(jx,jy));
}
}
}
// output << "RESULT X2: " << result(2,2,0) << endl;
if (mesh->ShiftXderivs)
result = result.shiftZ(false); // Back to field-aligned
// output << "RESULT X3: " << result(2,2,0) << endl;
Field3D dxf;
dxf.allocate();
for (int jx = mesh->xstart; jx <= mesh->xend; jx++) {
for (int jy = mesh->ystart; jy <= mesh->yend; jy++) {
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
int jxm = jx - 1;
BoutReal fac = 2.0;
if (mesh->firstX() && (jx == mesh->xstart)) {
jxm = jx;
// fac = 1.0;
}
int jxp = jx + 1;
if (mesh->lastX() && (jx == mesh->xend)) {
jxp = jx;
// fac = 1.0;
}
dxf(jx, jy, jz) =
(fs(jxp, jy, jz) - fs(jxm, jy, jz)) / (fac * mesh->dx(jx, jy));
}
}
}
if (mesh->ShiftXderivs)
dxf = dxf.shiftZ(false);
if (yderiv) {
// Y flux
Field3D Vy = a * (mesh->J * mesh->g_23 / mesh->g_22) * dxf;
// Vy.applyBoundary("neumann");
mesh->communicate(Vy);
for (int jx = mesh->xstart; jx <= mesh->xend; jx++) {
for (int jy = mesh->ystart; jy <= mesh->yend; jy++) {
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
// if(jx == mesh->xstart) {
// output.write("Y %d: + %e - %e\n", jy,
// n(jx,jy+1,jz)*Vy(jx,jy+1,jz), n(jx,jy-1,jz)*Vy(jx,jy-1,jz));
//}
// result(jx, jy, jz) += ( Vy(jx,jy,jz)*(n(jx,jy+1,jz) -
// n(jx,jy-1,jz)) + n(jx,jy,jz)*(Vy(jx,jy+1,jz) - Vy(jx,jy-1,jz)) ) /
// (2.*mesh->dy(jx, jy) * mesh->J(jx,jy));
result(jx, jy, jz) += (n(jx, jy + 1, jz) * Vy(jx, jy + 1, jz) -
n(jx, jy - 1, jz) * Vy(jx, jy - 1, jz)) /
(2. * mesh->dy(jx, jy) * mesh->J(jx, jy));
}
}
}
}
// output << "RESULT Y: " << result(2,2,0) << endl;
// Z flux
Field3D Vz = -a * mesh->J * dxf;
int ncz = mesh->ngz - 1;
for (int jx = mesh->xstart; jx <= mesh->xend; jx++) {
for (int jy = mesh->ystart; jy <= mesh->yend; jy++) {
for (int jz = 0; jz < ncz; jz++) {
// output.write("OLD: (%d,%d,%d) : (%e, %e)\n", jx,jy,jz,Vx(jx,jy,jz),
// Vz(jx,jy,jz));
int jzp = (jz + 1) % ncz;
int jzm = (jz - 1 + ncz) % ncz;
result(jx, jy, jz) += (n(jx, jy, jzp) * Vz(jx, jy, jzp) -
n(jx, jy, jzm) * Vz(jx, jy, jzm)) /
(2. * mesh->dz * mesh->J(jx, jy));
}
}
}
// output << "RESULT Z: " << result(2,2,0) << endl;
return result;
}
// Div ( n * b x Grad(f) / B ), where n is the quantity being advected
// This version uses central differences
const Field3D Div_n_bxGrad_f_B(const Field3D &n, const Field3D &f) {
Field3D result = 0.0;
// X flux
Field3D g = n * mesh->J * (mesh->g_23 / mesh->g_22);
Field3D Fx =
0.5 *
(
// n * a * mesh->J*(DDZ(f) - (mesh->g_23/mesh->g_22)*DDY(f,DIFF_C2)) +
-g * DDY(f, DIFF_C2) + f * DDY(g, DIFF_C2));
Fx.applyBoundary("neumann");
if (mesh->ShiftXderivs)
Fx = Fx.shiftZ(true); // Shift to orthogonal coordinates for X derivatives
for (int jx = mesh->xstart; jx <= mesh->xend; jx++) {
for (int jy = mesh->ystart; jy <= mesh->yend; jy++) {
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
result(jx, jy, jz) += (Fx(jx + 1, jy, jz) - Fx(jx - 1, jy, jz)) /
(2. * mesh->dx(jx, jy) * mesh->J(jx, jy));
}
}
}
if (mesh->ShiftXderivs)
result = result.shiftZ(false); // Back to field-aligned
// Y flux
Field3D Fy;
Fy.allocate();
for (int jx = mesh->xstart; jx <= mesh->xend; jx++) {
for (int jy = 0; jy < mesh->ngy; jy++) {
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
Fy(jx, jy, jz) =
0.5 * (g(jx, jy, jz) * (f(jx + 1, jy, jz) - f(jx - 1, jy, jz)) -
f(jx, jy, jz) * (g(jx + 1, jy, jz) - g(jx - 1, jy, jz))) /
(2. * mesh->dx(jx, jy));
}
}
}
// mesh->communicate(Vy);
// Fy.applyBoundary("neumann");
for (int jx = mesh->xstart; jx <= mesh->xend; jx++) {
for (int jy = mesh->ystart; jy <= mesh->yend; jy++) {
for (int jz = 0; jz < mesh->ngz - 1; jz++) {
result(jx, jy, jz) += (Fy(jx, jy + 1, jz) - Fy(jx, jy - 1, jz)) /
(2. * mesh->dy(jx, jy) * mesh->J(jx, jy));
}
}
}
/*
// Z flux
Field3D Vz = -DDX(f);
int ncz = mesh->ngz-1;
for(int jx=mesh->xstart; jx<=mesh->xend;jx++) {
for(int jy=mesh->ystart;jy<=mesh->yend;jy++) {
for(int jz=0;jz<ncz;jz++) {
// Flux at lower boundary
BoutReal fl = 0.5*(Fz(jx,jy,(jz-1+ncz)%ncz) + Fz(jx,jy,jz));
// Flux at upper boundary
BoutReal fu = 0.5*(Fz(jx,jy,jz) + Fz(jx,jy,(jz+1)%ncz));
result(jx, jy, jz) += (fu - fl) / mesh->dz;
//output.write("Val: %d,%d,%d : %e\n", jx,jy,jz, result(jx, jy, jz));
}
}
}
*/
return result;
}
// Div ( a * Grad_perp(f) )
const Field3D Div_Perp_Lap(const Field3D &a, const Field3D &fin) {
Field3D result;
result.allocate();
Field3D f = fin;
if (mesh->ShiftXderivs)
f = fin.shiftZ(true); // Shift so X-Z is orthogonal
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Z indices
int kp = (k + 1) % (mesh->ngz - 1);
int km = (k - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
int kpp = (kp + 1) % (mesh->ngz - 1);
int kmm = (km - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
BoutReal fm, fp;
// X flux
// Calculate derivatives at (i-1,j,k)
fm = mesh->g11(i - 1, j) * (f(i, j, k) - f(i - 2, j, k)) /
(2. * mesh->dx(i - 1, j))
//+ mesh->g12(i-1,j)*(f(i-1,j+1,k) -
//f(i-1,j-1,k))/(2.*mesh->dy(i-1,j))
+
mesh->g13(i - 1, j) * (f(i - 1, j, kp) - f(i - 1, j, km)) /
(2. * mesh->dz);
// Derivatives at (i,j,k)
// Derivatives at (i+1,j,k)
fp = mesh->g11(i + 1, j) * (f(i + 2, j, k) - f(i, j, k)) /
(2. * mesh->dx(i + 1, j))
//+ mesh->g12(i+1,j)*(f(i+1,j+1,k) -
//f(i+1,j-1,k))/(2.*mesh->dy(i-1,j))
+
mesh->g13(i + 1, j) * (f(i + 1, j, kp) - f(i + 1, j, km)) /
(2. * mesh->dz);
result(i, j, k) = (mesh->J(i + 1, j) * fp * a(i + 1, j, k) -
mesh->J(i - 1, j) * fm * a(i - 1, j, k)) /
(2. * mesh->J(i, j) * mesh->dx(i, j));
/*
if((i == 90) && (j == 8) && (k == 0)) {
output.write("Div: %e, %e, %e, %e, %e, %e\n",
f(i,j,k),
f(i,j,k) - f(i-2,j,k),
f(i+2,j,k) - f(i,j,k),
fm,
fp,
result(i,j,k));
}
*/
// Y flux
// Derivatives at (i,j-1,k)
fm = // mesh->g12(i,j-1)*(f(i+1,j-1,k) -
// f(i-1,j-1,k))/(2.*mesh->dx(i,j-1))
+(mesh->g22(i, j - 1) - 1. / mesh->g_22(i, j - 1)) *
(f(i, j + 1, k) - f(i, j - 1, k)) / (2. * mesh->dy(i, j - 1)) +
mesh->g23(i, j - 1) * (f(i, j - 1, kp) - f(i, j - 1, km)) /
(2. * mesh->dz);
// Derivatives at (i,j+1,k)
fp = // mesh->g12(i,j+1)*(f(i+1,j-1,k) -
// f(i-1,j-1,k))/(2.*mesh->dx(i,j-1))
+(mesh->g22(i, j + 1) - 1. / mesh->g_22(i, j + 1)) *
(f(i, j + 2, k) - f(i, j, k)) / (2. * mesh->dy(i, j + 1)) +
mesh->g23(i, j + 1) * (f(i, j + 1, kp) - f(i, j + 1, km)) /
(2. * mesh->dz);
// result(i,j,k) += (mesh->J(i,j+1)*fp*a(i,j+1,k) -
// mesh->J(i,j-1)*fm*a(i,j-1,k))/(mesh->J(i,j)*mesh->dy(i,j));
// Z flux
// Derivatives at (i,j,k-1)
fm = mesh->g13(i, j) * (f(i + 1, j, km) - f(i - 1, j, km)) /
(2. * mesh->dx(i, j)) +
mesh->g23(i, j) * (f(i, j + 1, km) - f(i, j - 1, km)) /
(2. * mesh->dy(i, j)) +
mesh->g33(i, j) * (f(i, j, k) - f(i, j, kmm)) / (2. * mesh->dz);
fp = mesh->g13(i, j) * (f(i + 1, j, kp) - f(i - 1, j, kp)) /
(2. * mesh->dx(i, j)) +
mesh->g23(i, j) * (f(i, j + 1, kp) - f(i, j - 1, kp)) /
(2. * mesh->dy(i, j)) +
mesh->g33(i, j) * (f(i, j, kpp) - f(i, j, k)) / (2. * mesh->dz);
// result(i,j,k) += (fp*a(i,j,k+1) - fm*a(i,j,k-1))/(2.*mesh->dz);
}
return result;
}
// Div ( a * Grad_perp(f) )
const Field3D Div_Perp_Lap_x3(const Field3D &a, const Field3D &f, bool xflux) {
Field3D result;
result.allocate();
Field3D fs = f;
Field3D as = a;
if (mesh->ShiftXderivs) {
fs = f.shiftZ(true);
as = a.shiftZ(true);
}
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Z indices
int kp = (k + 1) % (mesh->ngz - 1);
int km = (k - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
BoutReal fm, fp;
// X flux
// Calculate derivatives at (i-1/2,j,k)
fm = 0.5 * (mesh->g11(i - 1, j) + mesh->g11(i, j)) *
(fs(i, j, k) - fs(i - 1, j, k)) /
(0.5 * (mesh->dx(i - 1, j) + mesh->dx(i, j)))
//+ mesh->g12(i-1,j)*(f(i-1,j+1,k) -
//f(i-1,j-1,k))/(2.*mesh->dy(i-1,j))
+
0.5 * (mesh->g13(i - 1, j) + mesh->g13(i, j)) *
(fs(i - 1, j, kp) - fs(i - 1, j, km) + f(i, j, kp) -
f(i, j, km)) /
(4. * mesh->dz);
// Derivatives at (i+1/2,j,k)
fp = 0.5 * (mesh->g11(i, j) + mesh->g11(i + 1, j)) *
(fs(i + 1, j, k) - fs(i, j, k)) /
(0.5 * (mesh->dx(i + 1, j) + mesh->dx(i, j)))
//+ mesh->g12(i+1,j)*(f(i+1,j+1,k) -
//f(i+1,j-1,k))/(2.*mesh->dy(i-1,j))
+
0.5 * (mesh->g13(i, j) + mesh->g13(i + 1, j)) *
(fs(i + 1, j, kp) - fs(i + 1, j, km) + fs(i, j, kp) -
fs(i, j, km)) /
(4. * mesh->dz);
if (mesh->firstX() && (i == mesh->xstart) && (!xflux))
fm = 0.0;
if (mesh->lastX() && (i == mesh->xend) && (!xflux))
fp = 0.0;
result(i, j, k) = (+fp * 0.5 * (mesh->J(i + 1, j) * as(i + 1, j, k) +
mesh->J(i, j) * as(i, j, k)) -
fm * 0.5 * (mesh->J(i - 1, j) * as(i - 1, j, k) +
mesh->J(i, j) * as(i, j, k))) /
(mesh->J(i, j) * mesh->dx(i, j));
/*
// Y flux
// Derivatives at (i,j-1,k)
fm = //mesh->g12(i,j-1)*(f(i+1,j-1,k) -
f(i-1,j-1,k))/(2.*mesh->dx(i,j-1))
+ (mesh->g22(i,j-1) - 1./mesh->g_22(i,j-1))*(f(i,j+1,k) -
f(i,j-1,k))/(2.*mesh->dy(i,j-1))
+ mesh->g23(i,j-1)*(f(i,j-1,kp) - f(i,j-1,km))/(2.*mesh->dz);
// Derivatives at (i,j+1,k)
fp = //mesh->g12(i,j+1)*(f(i+1,j-1,k) -
f(i-1,j-1,k))/(2.*mesh->dx(i,j-1))
+ (mesh->g22(i,j+1) - 1./mesh->g_22(i,j+1))*(f(i,j+2,k) -
f(i,j,k))/(2.*mesh->dy(i,j+1))
+ mesh->g23(i,j+1)*(f(i,j+1,kp) - f(i,j+1,km))/(2.*mesh->dz);
//result(i,j,k) += (mesh->J(i,j+1)*fp*a(i,j+1,k) -
mesh->J(i,j-1)*fm*a(i,j-1,k))/(mesh->J(i,j)*mesh->dy(i,j));
*/
// Z flux
// Derivatives at (i,j,k-1)
fm = mesh->g13(i, j) * (fs(i + 1, j, km) - fs(i - 1, j, km)) /
(2. * mesh->dx(i, j))
//+ mesh->g23(i,j)*(f(i,j+1,km) - f(i,j-1,km))/(2.*mesh->dy(i,j))
+ mesh->g33(i, j) * (fs(i, j, k) - fs(i, j, km)) / (2. * mesh->dz);
fp = mesh->g13(i, j) * (fs(i + 1, j, kp) - fs(i - 1, j, kp)) /
(2. * mesh->dx(i, j))
//+ mesh->g23(i,j)*(f(i,j+1,kp) - f(i,j-1,kp))/(2.*mesh->dy(i,j))
+ mesh->g33(i, j) * (fs(i, j, kp) - fs(i, j, k)) / (2. * mesh->dz);
result(i, j, k) +=
(fp * a(i, j, kp) - fm * a(i, j, km)) / (2. * mesh->dz);
}
if (mesh->ShiftXderivs)
result = result.shiftZ(false);
return result;
}
// Div ( a * Grad_perp(f) )
const Field3D Div_Perp_Lap_XYZ(const Field3D &a, const Field3D &f,
bool bndryflux) {
Field3D result = 0.0;
Field3D fs = f;
Field3D as = a;
if (mesh->ShiftXderivs) {
fs = f.shiftZ(true);
as = a.shiftZ(true);
}
// X flux
int xfirst = mesh->xstart;
int xlast = mesh->xend + 1;
if (!bndryflux && !mesh->periodicX) {
if (mesh->firstX())
++xfirst;
if (mesh->lastX())
--xlast;
}
for (int i = xfirst; i <= xlast; i++)
for (int j = mesh->ystart; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Z indices
int kp = (k + 1) % (mesh->ngz - 1);
int km = (k - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
// X flux
// Calculate derivatives at (i-1/2,j,k)
BoutReal fm = 0.5 * (mesh->g11(i - 1, j) + mesh->g11(i, j)) *
(fs(i, j, k) - fs(i - 1, j, k)) /
(0.5 * (mesh->dx(i - 1, j) + mesh->dx(i, j)))
//+ mesh->g12(i-1,j)*(f(i-1,j+1,k) -
//f(i-1,j-1,k))/(2.*mesh->dy(i-1,j))
//+ 0.5*(mesh->g13(i-1,j) + mesh->g13(i,j))*(fs(i-1,j,kp) -
//fs(i-1,j,km) + f(i,j,kp) - f(i,j,km))/(4.*mesh->dz)
;
BoutReal flux = -fm * 0.5 * (mesh->J(i - 1, j) + mesh->J(i, j)) * 0.5 *
(as(i - 1, j, k) + as(i, j, k));
result(i, j, k) += flux / (mesh->J(i, j) * mesh->dx(i, j));
result(i - 1, j, k) -= flux / (mesh->J(i - 1, j) * mesh->dx(i - 1, j));
}
if (mesh->ShiftXderivs)
result = result.shiftZ(false);
// Y flux
for (int i = mesh->xstart; i <= mesh->xend; i++) {
int yfirst = mesh->ystart;
int ylast = mesh->yend + 1;
if (!bndryflux && !mesh->periodicY(i)) {
if (mesh->firstY(i))
++yfirst;
if (mesh->lastY(i))
--ylast;
}
for (int j = yfirst; j <= ylast; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Z indices
int kp = (k + 1) % (mesh->ngz - 1);
int km = (k - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
// Derivatives at (i,j-1/2,k)
BoutReal fm = // mesh->g12(i,j-1)*(f(i+1,j-1,k) -
// f(i-1,j-1,k))/(2.*mesh->dx(i,j-1))
+(mesh->g22(i, j - 1) + mesh->g22(i, j) -
1. / mesh->g_22(i, j - 1) - 1. / mesh->g_22(i, j)) *
(f(i, j, k) - f(i, j - 1, k)) /
(mesh->dy(i, j - 1) + mesh->dy(i, j)) +
0.5 * (mesh->g23(i, j - 1) + mesh->g23(i, j)) *
(f(i, j - 1, kp) + f(i, j, kp) - f(i, j - 1, km) -
f(i, j, km)) /
(4. * mesh->dz);
BoutReal flux = -fm * 0.25 * (mesh->J(i, j - 1) + mesh->J(i, j)) *
(a(i, j - 1, k) + a(i, j, k));
result(i, j, k) += flux / (mesh->J(i, j) * mesh->dy(i, j));
result(i, j - 1, k) -= flux / (mesh->J(i, j - 1) * mesh->dy(i, j - 1));
// if(i == 30) {
// output.write("%d: %e, %e, %e, %e\n", j, fm, flux, flux /
// (mesh->J(i,j)*mesh->dy(i,j)), flux /
// (mesh->J(i,j-1)*mesh->dy(i,j-1)));
//}
}
}
// Z flux
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Z indices
int kp = (k + 1) % (mesh->ngz - 1);
int km = (k - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
// Derivatives at (i,j,k-1)
BoutReal fm = // mesh->g13(i,j)*(fs(i+1,j,km) -
// fs(i-1,j,km))/(2.*mesh->dx(i,j))
+mesh->g23(i, j) * ((f(i, j + 1, km) + f(i, j + 1, k)) -
(f(i, j - 1, km) + f(i, j - 1, k))) /
(4. * mesh->dy(i, j)) +
mesh->g33(i, j) * (f(i, j, k) - f(i, j, km)) / mesh->dz;
BoutReal flux = -fm * 0.5 * (a(i, j, km) + a(i, j, k)) / mesh->dz;
result(i, j, k) += flux;
result(i, j, km) -= flux;
}
return result;
}
const Field3D Div_par_diffusion(const Field3D &K, const Field3D &f,
bool bndry_flux) {
Field3D result;
result = 0.0;
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart - 1; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Calculate flux at upper surface
if (!bndry_flux && !mesh->periodicY(i)) {
if ((j == mesh->yend) && mesh->lastY(i))
continue;
if ((j == mesh->ystart - 1) && mesh->firstY(i))
continue;
}
BoutReal c =
0.5 * (K(i, j, k) + K(i, j + 1, k)); // K at the upper boundary
BoutReal J =
0.5 * (mesh->J(i, j) + mesh->J(i, j + 1)); // Jacobian at boundary
// if((i == mesh->xstart) && (j == mesh->ystart) && (k==0))
// output << "c = " << c << endl;
BoutReal g_22 = 0.5 * (mesh->g_22(i, j) + mesh->g_22(i, j + 1));
BoutReal gradient = 2. * (f(i, j + 1, k) - f(i, j, k)) /
(mesh->dy(i, j) + mesh->dy(i, j + 1));
BoutReal flux = c * J * gradient / g_22;
result(i, j, k) += flux / (mesh->dy(i, j) * mesh->J(i, j));
result(i, j + 1, k) -= flux / (mesh->dy(i, j + 1) * mesh->J(i, j + 1));
}
return result;
}
const Field3D Div_par_spitzer(BoutReal K0, const Field3D &Te, bool bndry_flux) {
Field3D result;
result = 0.0;
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart - 1; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Calculate flux at upper surface
if (!bndry_flux && !mesh->periodicY(i)) {
if ((j == mesh->yend) && mesh->lastY(i))
continue;
if ((j == mesh->ystart - 1) && mesh->firstY(i))
continue;
}
BoutReal Te0 =
0.5 * (Te(i, j, k) + Te(i, j + 1, k)); // Te at the upper boundary
BoutReal K = K0 * pow(Te0, 2.5);
BoutReal J =
0.5 * (mesh->J(i, j) + mesh->J(i, j + 1)); // Jacobian at boundary
BoutReal g_22 = 0.5 * (mesh->g_22(i, j) + mesh->g_22(i, j + 1));
BoutReal gradient = 2. * (Te(i, j + 1, k) - Te(i, j, k)) /
(mesh->dy(i, j) + mesh->dy(i, j + 1));
BoutReal flux = K * J * gradient / g_22;
result(i, j, k) += flux / (mesh->dy(i, j) * mesh->J(i, j));
result(i, j + 1, k) -= flux / (mesh->dy(i, j + 1) * mesh->J(i, j + 1));
}
return result;
}
const Field3D Div_par_diffusion_upwind(const Field3D &K, const Field3D &f,
bool bndry_flux) {
Field3D result;
result = 0.0;
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart - 1; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Calculate flux at upper surface
if (!bndry_flux && !mesh->periodicY(i)) {
if ((j == mesh->yend) && mesh->lastY(i))
continue;
if ((j == mesh->ystart - 1) && mesh->firstY(i))
continue;
}
BoutReal J =
0.5 * (mesh->J(i, j) + mesh->J(i, j + 1)); // Jacobian at boundary
BoutReal g_22 = 0.5 * (mesh->g_22(i, j) + mesh->g_22(i, j + 1));
BoutReal gradient = 2. * (f(i, j + 1, k) - f(i, j, k)) /
(mesh->dy(i, j) + mesh->dy(i, j + 1));
BoutReal c; // K at the upper boundary
if (gradient > 0.0) {
c = K(i, j + 1, k);
} else {
c = K(i, j, k);
}
BoutReal flux = c * J * gradient / g_22;
result(i, j, k) += flux / (mesh->dy(i, j) * mesh->J(i, j));
result(i, j + 1, k) -= flux / (mesh->dy(i, j + 1) * mesh->J(i, j + 1));
}
return result;
}
const Field3D Div_par_diffusion_index(const Field3D &f, bool bndry_flux) {
Field3D result;
result = 0.0;
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart - 1; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Calculate flux at upper surface
if (!bndry_flux && !mesh->periodicY(i)) {
if ((j == mesh->yend) && mesh->lastY(i))
continue;
if ((j == mesh->ystart - 1) && mesh->firstY(i))
continue;
}
BoutReal J =
0.5 * (mesh->J(i, j) + mesh->J(i, j + 1)); // Jacobian at boundary
BoutReal gradient = f(i, j + 1, k) - f(i, j, k);
BoutReal flux = J * gradient;
result(i, j, k) += flux / mesh->J(i, j);
result(i, j + 1, k) -= flux / mesh->J(i, j + 1);
}
return result;
}
/*
* Added Dissipation scheme (related to Momentum Interpolation)
*
* This uses a 3rd-order derivative of the pressure as
* a correction to the velocity.
*
* This should appear in the form
*
* df/dt = ... + AddedDissipation(N, P, f);
*/
const Field3D AddedDissipation(const Field3D &N, const Field3D &P,
const Field3D f, bool bndry_flux) {
Field3D result = 0.0;
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart - 1; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
// Calculate flux at upper surface
if (!bndry_flux && !mesh->periodicY(i)) {
if ((j >= mesh->yend) && mesh->lastY(i))
continue;
if ((j <= mesh->ystart) && mesh->firstY(i))
continue;
}
// At upper boundary
BoutReal nbndry = 0.5 * N(i, j, k) + N(i, j + 1, k);
if (nbndry < 1e-5)
nbndry = 1e-5;
BoutReal d = 1. / nbndry; // 0.5*(1./N(i,j,k) + 1./N(i,j+1,k));
// Velocity
BoutReal v =
-0.25 * d * ((P(i, j - 1, k) + P(i, j + 1, k) - 2. * P(i, j, k)) -
(P(i, j, k) + P(i, j + 2, k) - 2. * P(i, j + 1, k)));
// Variable being advected. Could use different interpolation?
BoutReal var = 0.5 * (f(i, j, k) + f(i, j + 1, k));
BoutReal flux = var * v * (mesh->J(i, j) + mesh->J(i, j + 1)) /
(sqrt(mesh->g_22(i, j)) + sqrt(mesh->g_22(i, j + 1)));
result(i, j, k) -= flux / (mesh->dy(i, j) * mesh->J(i, j));
result(i, j + 1, k) += flux / (mesh->dy(i, j + 1) * mesh->J(i, j + 1));
}
return result;
}
////////////////////////////////////////////////////////////////////////////////////////////////
// XPPM methods
BoutReal BOUTMIN(const BoutReal &a, const BoutReal &b, const BoutReal &c,
const BoutReal &d) {
BoutReal r1 = (a < b) ? a : b;
BoutReal r2 = (c < d) ? c : d;
return (r1 < r2) ? r1 : r2;
}
struct Stencil1D {
// Cell centre values
BoutReal c, m, p, mm, pp;
// Left and right cell face values
BoutReal L, R;
};
// First order upwind for testing
void Upwind(Stencil1D &n, const BoutReal h) { n.L = n.R = n.c; }
// Fromm method
void Fromm(Stencil1D &n, const BoutReal h) {
n.L = n.c - 0.25 * (n.p - n.m);
n.R = n.c + 0.25 * (n.p - n.m);
}
/// The minmod function returns the value with the minimum magnitude
/// If the inputs have different signs then returns zero
BoutReal minmod(BoutReal a, BoutReal b) {
if (a * b <= 0.0)
return 0.0;
if (fabs(a) < fabs(b))
return a;
return b;
}
BoutReal minmod(BoutReal a, BoutReal b, BoutReal c) {
// If any of the signs are different, return zero gradient
if ((a * b <= 0.0) || (a * c <= 0.0)) {
return 0.0;
}
// Return the minimum absolute value
return SIGN(a) * BOUTMIN(fabs(a), fabs(b), fabs(c));
}
void MinMod(Stencil1D &n, const BoutReal h) {
// Choose the gradient within the cell
// as the minimum (smoothest) solution
BoutReal slope = minmod(n.p - n.c, n.c - n.m);
n.L = n.c - 0.5 * slope; // 0.25*(n.p - n.m);
n.R = n.c + 0.5 * slope; // 0.25*(n.p - n.m);
}
// Monotonized Central limiter (Van-Leer)
void MC(Stencil1D &n, const BoutReal h) {
BoutReal slope =
minmod(2. * (n.p - n.c), 0.5 * (n.p - n.m), 2. * (n.c - n.m));
n.L = n.c - 0.5 * slope;
n.R = n.c + 0.5 * slope;
}
void XPPM(Stencil1D &n, const BoutReal h) {
// 4th-order PPM interpolation in X
const BoutReal C = 1.25; // Limiter parameter
BoutReal h2 = h * h;
n.R = (7. / 12) * (n.c + n.p) - (1. / 12) * (n.m + n.pp);
n.L = (7. / 12) * (n.c + n.m) - (1. / 12) * (n.mm + n.p);
// Apply limiters
if ((n.c - n.R) * (n.p - n.R) > 0.0) {
// Calculate approximations to second derivative
BoutReal D2 = (3. / h2) * (n.c - 2 * n.R + n.p);
BoutReal D2L = (1. / h2) * (n.m - 2 * n.c + n.p);
BoutReal D2R = (1. / h2) * (n.c - 2. * n.p + n.pp);
BoutReal D2lim; // Value to be used in limiter
// Check if they all have the same sign
if ((D2 * D2L > 0.0) && (D2 * D2R > 0.0)) {
// Same sign
D2lim = SIGN(D2) * BOUTMIN(C * fabs(D2L), C * fabs(D2R), fabs(D2));
} else {
// Different sign
D2lim = 0.0;
}
n.R = 0.5 * (n.c + n.p) - (h2 / 6) * D2lim;
}
if ((n.m - n.L) * (n.c - n.L) > 0.0) {
// Calculate approximations to second derivative
BoutReal D2 = (3. / h2) * (n.m - 2 * n.L + n.c);
BoutReal D2L = (1. / h2) * (n.mm - 2 * n.m + n.c);
BoutReal D2R = (1. / h2) * (n.m - 2. * n.c + n.p);
BoutReal D2lim; // Value to be used in limiter
// Check if they all have the same sign
if ((D2 * D2L > 0.0) && (D2 * D2R > 0.0)) {
// Same sign
D2lim = SIGN(D2) * BOUTMIN(C * fabs(D2L), C * fabs(D2R), fabs(D2));
} else {
// Different sign
D2lim = 0.0;
}
n.L = 0.5 * (n.m + n.c) - (h2 / 6) * D2lim;
}
if (((n.R - n.c) * (n.c - n.L) <= 0.0) ||
((n.m - n.c) * (n.c - n.p) <= 0.0)) {
// At a local maximum or minimum
BoutReal D2 = (6. / h2) * (n.L - 2. * n.c + n.R);
if (fabs(D2) < 1e-10) {
n.R = n.L = n.c;
} else {
BoutReal D2C = (1. / h2) * (n.m - 2. * n.c + n.p);
BoutReal D2L = (1. / h2) * (n.mm - 2 * n.m + n.c);
BoutReal D2R = (1. / h2) * (n.c - 2. * n.p + n.pp);
BoutReal D2lim;
// Check if they all have the same sign
if ((D2 * D2C > 0.0) && (D2 * D2L > 0.0) && (D2 * D2R > 0.0)) {
// Same sign
D2lim = SIGN(D2) *
BOUTMIN(C * fabs(D2L), C * fabs(D2R), C * fabs(D2C), fabs(D2));
n.R = n.c + (n.R - n.c) * D2lim / D2;
n.L = n.c + (n.L - n.c) * D2lim / D2;
} else {
// Different signs
n.R = n.L = n.c;
}
}
}
}
/*
* Div (n * b x Grad(f)/B)
*
*
* poloidal - If true, includes X-Y flows
* positive - If true, limit advected quantity (n_in) to be positive
*/
const Field3D Div_n_bxGrad_f_B_XPPM(const Field3D &n_in, const Field3D &f_in,
bool bndry_flux, bool poloidal,
bool positive) {
Field3D result = 0;
//////////////////////////////////////////
// X-Z advection.
//
// Z
// |
//
// fmp --- vU --- fpp
// | nU |
// | |
// vL nL nR vR -> X
// | |
// | nD |
// fmm --- vD --- fpm
//
Field3D n = n_in; // Done in orthogonal X-Z coordinates
Field3D f = f_in;
if (mesh->ShiftXderivs) {
n = n_in.shiftZ(true); // Done in orthogonal X-Z coordinates
f = f_in.shiftZ(true);
}
for (int i = mesh->xstart; i <= mesh->xend; i++)
for (int j = mesh->ystart; j <= mesh->yend; j++)
for (int k = 0; k < mesh->ngz - 1; k++) {
int kp = (k + 1) % (mesh->ngz - 1);
int kpp = (kp + 1) % (mesh->ngz - 1);
int km = (k - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
int kmm = (km - 1 + mesh->ngz - 1) % (mesh->ngz - 1);
// 1) Interpolate stream function f onto corners fmp, fpp, fpm
BoutReal fmm = 0.25 * (f(i, j, k) + f(i - 1, j, k) + f(i, j, km) +
f(i - 1, j, km));
BoutReal fmp = 0.25 * (f(i, j, k) + f(i, j, kp) + f(i - 1, j, k) +
f(i - 1, j, kp)); // 2nd order accurate
BoutReal fpp = 0.25 * (f(i, j, k) + f(i, j, kp) + f(i + 1, j, k) +