Equations

Fluids, radiation, magnetic fields, and dust

Assuming the speed of light is not reduced ($\hat{c} = c$), Quokka solves the following conservation laws for the cell-centered variables, as written below in the Heaviside–Lorentz system of units:

$\frac{\partial \vec{U}}{\partial t}+\nabla \cdot \vec{F}(\vec{U}) = \vec{S}(\vec{U}),$ $\begin{aligned} \vec{U} =\left[ \begin{array}{c} \rho \\ \rho \vec{v} \\ E \\ \rho X_n \\ E_g \\ \vec{F}_g \\ \rho_{\mathrm{d},k} \\ \rho_{\mathrm{d},k} \vec{v}_{\mathrm{d},k} \end{array}\right], \; \vec{F}(U) = \left[ \begin{array}{c} \rho \vec{v} \\ \rho \vec{v} \otimes \vec{v} + \left(p + \frac{1}{2} B^2\right)\mathsf{I} - \vec{B} \otimes \vec{B} \\ \left(E + p + \frac{1}{2} B^2\right)\vec{v} - (\vec{v} \cdot \vec{B}) \vec{B} \\ \rho X_n \vec{v} \\ \vec{F}_g \\ c^2 \mathsf{P}_g \\ \rho_{\mathrm{d},k} \vec{v}_{\mathrm{d},k} \\ \rho_{\mathrm{d},k} \vec{v}_{\mathrm{d},k} \otimes \vec{v}_{\mathrm{d},k} \end{array}\right], \; \vec{S}(U)=\left[ \begin{array}{c} 0 \\ \sum_g \vec{G}_g + \rho \vec{g} + \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{\vec{v}_{\mathrm{d},k} - \vec{v}}{T_{\mathrm{s},k}} \\ c \sum_g G^0_{g} + \rho \vec{v} \cdot \vec{g} + \mathcal{H} - \mathcal{C} + \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{\vec{v}_{\mathrm{d},k} - \vec{v}}{T_{\mathrm{s},k}} \cdot \vec{v} + \omega \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{(\vec{v}_{\mathrm{d},k} - \vec{v})^{2}}{T_{\mathrm{s},k}} \\ \rho \dot{X}_n \\ - c G^0_{g} \\ - c^2 \vec{G}_g \\ 0 \\ \rho_{\mathrm{d},k} \frac{\vec{v} - \vec{v}_{\mathrm{d},k}}{T_{\mathrm{s},k}} \end{array}\right], \end{aligned}$

where the total fluid energy is

$E = \rho e + \frac{1}{2} \rho v^2 + \frac{1}{2} B^2 \, ,$

and the face-centered magnetic field is evolved according to the ideal MHD induction equation:

$\frac{\partial \vec{B}}{\partial t} - \nabla \times \left(\vec{v} \times \vec{B}\right) = 0.$

Quokka also solves the non-conservative auxiliary internal energy equation:

$\begin{aligned} \frac{\partial (\rho e_{\text{aux}})}{\partial t} = - \nabla \cdot (\rho e_{\text{aux}} \vec{v}) - p \nabla \cdot \vec{v} + S_{\text{rad}} + \mathcal{H} - \mathcal{C} + \omega \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{(\vec{v}_{\mathrm{d},k} - \vec{v})^{2}}{T_{\mathrm{s},k}}, \\ \Delta S_{\text{rad}} = \int \sum_g c G^0_g \ dt - \frac{1}{2} \Delta \left(\rho v^2 \right), \end{aligned}$

and the gravitational Poisson equation:

$\begin{aligned} \nabla^2 \phi = -4 \pi G \left( \rho + \sum_i \rho_i \right), \\ \vec{g} \equiv -\nabla \phi, \end{aligned}$

where

$\rho$ is the gas density,
$\vec{v}$ is the gas velocity,
$E$ is the total fluid energy density, including magnetic energy when MHD is enabled,
$\rho e_{\text{aux}}$ is the auxiliary gas internal energy density,
$X_n$ is the fractional concentration of species $n$,
$\dot{X}_n$ is the chemical reaction term for species $n$,
$\mathcal{H}$ is the optically-thin volumetric heating term (radiative and chemical),
$\mathcal{C}$ is the optically-thin volumetric cooling term (radiative and chemical),
$p(\rho, e)$ is the gas pressure derived from a general convex equation of state,
$\vec{B}$ is the magnetic field,
$\mathsf{I}$ is the identity tensor,
$E_g$ is the radiation energy density for group $g$,
$F_g$ is the radiation flux for group $g$,
$\mathsf{P}_g$ is the radiation pressure tensor for group $g$,
$G_g$ is the radiation four-force $[G^0_g, \vec{G}_g]$ due to group $g$,
$\Delta S_{\text{rad}}$ is the change in gas internal energy due to radiation over a timestep,
$\phi$ is the Newtonian gravitational potential,
$\vec{g}$ is the gravitational acceleration,
$\rho_i$ is the mass density due to particle $i$,
$\rho_{\mathrm{d},k}$ is the dust mass density for dust species $k$ ($k \in [1, N_{\mathrm{dust}}]$),
$\vec{v}_{\mathrm{d},k}$ is the dust velocity for dust species $k$,
$T_{\mathrm{s},k}$ is the aerodynamic stopping time for dust species $k$,
$\omega$ is the fraction of frictional heating deposited into the gas.

Note that since work done by radiation on the gas is included in the $c \sum_g G^0_g$ term, $S_{\text{rad}}$ is not the same as $c \sum_g G^0_g$.

Collisionless particles

Quokka solves the following equation of motion for collisionless particles:

$\frac{d^2 \vec{x}_i}{d t^2} = \vec{g} ,$

where $\vec{x}_i$ is the position vector of particle $i$.

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Quokka

Equations

Fluids, radiation, magnetic fields, and dust

Collisionless particles