add equations to docs (#691)

### Description
This adds the complete set of equations that are solved by Quokka to the
documentation.

### Related issues
N/A

### Checklist
_Before this pull request can be reviewed, all of these tasks should be
completed. Denote completed tasks with an `x` inside the square brackets
`[ ]` in the Markdown source below:_
- [x] I have added a description (see above).
- [ ] I have added a link to any related issues see (see above).
- [x] I have read the [Contributing
Guide](https://github.com/quokka-astro/quokka/blob/development/CONTRIBUTING.md).
- [ ] I have added tests for any new physics that this PR adds to the
code.
- [ ] I have tested this PR on my local computer and all tests pass.
- [ ] I have manually triggered the GPU tests with the magic comment
`/azp run`.
- [x] I have requested a reviewer for this PR.

docs/equations.rst

@@ -3,14 +3,102 @@
 Equations
 ==========================
 
-*This page is still under construction.*
+Fluids and radiation
+-----------------
+
+Assuming the speed of light is not reduced (:math:`\hat{c} = c`),
+Quokka solves the system of conservation laws:
+
+.. math::
+
+    \frac{\partial \vec{U}}{\partial t}+\nabla \cdot \vec{F}(\vec{U}) = \vec{S}(\vec{U}),
+
+..
+
+.. math::
+
+    \vec{U} =\left[
+    \begin{array}{c}
+      \rho \\
+      \rho \vec{v} \\
+      E_{\rm gas} \\
+      \rho X_n \\
+      E_g \\
+      \vec{F}_g
+    \end{array}\right], \;
+    \vec{F}(U) = \left[
+    \begin{array}{c}
+      \rho \vec{v} \\
+      \rho \vec{v} \otimes \vec{v}+p \\
+      (E_{\rm gas} + p) \vec{v} \\
+      \rho X_n \vec{v} \\
+      \vec{F}_g \\
+      c^2 \boldsymbol{P}_g
+    \end{array}\right], \;
+    \vec{S}(U)=\left[
+    \begin{array}{c}
+      0 \\
+      \sum_g \vec{G}_g + \rho \vec{g} \\
+      c \sum_g G^0_{g} + \rho \vec{v} \cdot \vec{g} + \mathcal{H} - \mathcal{C} \\
+      \rho \dot{X}_n \\
+      - c G^0_{g} \\
+      - c^2 \vec{G}_g
+    \end{array}\right],
+
+..
+
+along with the non-conservative auxiliary internal energy equation:
 
 .. math::
 
-    \frac{\partial E}{\partial t} + \nabla \cdot \vec{F} = -G^0 \\
-    \frac{1}{c^2} \frac{\partial F}{\partial t} + \nabla \cdot \vec{P} = -\vec{G}
+    \frac{\partial (\rho e_{\text{aux}})}{\partial t} =
+    - \nabla \cdot (\rho e_{\text{aux}} \vec{v}) - p \nabla \cdot \vec{v}
+    + S_{\text{rad}} + \mathcal{H} - \mathcal{C}, \\
+    \Delta S_{\text{rad}} = \int \sum_g c G^0_g \ dt - \frac{1}{2} \Delta \left(\rho v^2 \right),
 ..
 
-where :math:`E` is the radiation energy density, :math:`F` is the radiation flux,
-:math:`P` is the radiation pressure tensor, and :math:`G` is the radiation four-force.
+and the gravitational Poisson equation:
+
+.. math::
+
+    \nabla^2 \phi = -4 \pi G \left( \rho + \sum_i \rho_i \right), \\
+    \vec{g} \equiv -\nabla \phi,
+
+..
+
+where
+
+* :math:`\rho` is the gas density,
+* :math:`\vec{v}` is the gas velocity,
+* :math:`E_{\text{gas}}` is the total gas energy,
+* :math:`\rho e_{\text{aux}}` is the auxiliary gas internal energy,
+* :math:`X_n` is the fractional concentration of species :math:`n`,
+* :math:`\dot{X}_n` is the chemical reaction term for species :math:`n`,
+* :math:`\mathcal{H}` is the optically-thin volumetric heating term (radiative and chemical),
+* :math:`\mathcal{C}` is the optically-thin volumetric cooling term (radiative and chemical),
+* :math:`p(\rho, e)` is the gas pressure derived from a general convex equation of state,
+* :math:`E_g` is the radiation energy density for group :math:`g`,
+* :math:`F_g` is the radiation flux for group :math:`g`,
+* :math:`\boldsymbol{P}_g` is the radiation pressure tensor for group :math:`g`,
+* :math:`G_g` is the radiation four-force :math:`[G^0_g, \vec{G}_g]` due to group :math:`g`,
+* :math:`\Delta S_{\text{rad}}` is the change in gas internal energy due to radiation over a timestep,
+* :math:`\phi` is the Newtonian gravitational potential,
+* :math:`\vec{g}` is the gravitational acceleration,
+* :math:`\rho_i` is the mass density due to particle :math:`i`.
+
+Note that since work done by radiation on the gas is included in the
+:math:`c \sum_g G^0_g` term, :math:`S_{\text{rad}}` is not the same as
+:math:`c \sum_g G^0_g`.
+
+Collisionless particles
+-----------------------
+
+Quokka solves the following equation of motion for collisionless particles:
+
+.. math::
+
+    \frac{d^2 \vec{x}_i}{d t^2} = \vec{g} ,
+
+..
 
+where :math:`\vec{x}_i` is the position vector of particle :math:`i`.