Equations

Fluids and radiation

Assuming the speed of light is not reduced (\(\hat{c} = c\)), Quokka solves the system of conservation laws:

\[\frac{\partial \vec{U}}{\partial t}+\nabla \cdot \vec{F}(\vec{U}) = \vec{S}(\vec{U}),\]

\[\begin{aligned} \vec{U} =\left[ \begin{array}{c} \rho \\ \rho \vec{v} \\ E_{\rm gas} \\ \rho X_n \\ E_g \\ \vec{F}_g \end{array}\right], \; \vec{F}(U) = \left[ \begin{array}{c} \rho \vec{v} \\ \rho \vec{v} \otimes \vec{v}+p \\ (E_{\rm gas} + p) \vec{v} \\ \rho X_n \vec{v} \\ \vec{F}_g \\ c^2 \mathsf{P}_g \end{array}\right], \; \vec{S}(U)=\left[ \begin{array}{c} 0 \\ \sum_g \vec{G}_g + \rho \vec{g} \\ c \sum_g G^0_{g} + \rho \vec{v} \cdot \vec{g} + \mathcal{H} - \mathcal{C} \\ \rho \dot{X}_n \\ - c G^0_{g} \\ - c^2 \vec{G}_g \end{array}\right], \end{aligned}\]

along with the non-conservative auxiliary internal energy equation:

\[\begin{aligned} \frac{\partial (\rho e_{\text{aux}})}{\partial t} = - \nabla \cdot (\rho e_{\text{aux}} \vec{v}) - p \nabla \cdot \vec{v} + S_{\text{rad}} + \mathcal{H} - \mathcal{C}, \\ \Delta S_{\text{rad}} = \int \sum_g c G^0_g \ dt - \frac{1}{2} \Delta \left(\rho v^2 \right), \end{aligned}\]

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,
\(E_{\text{gas}}\) is the total gas energy,
\(\rho e_{\text{aux}}\) is the auxiliary gas internal energy,
\(X_n\) 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,
\(E_g\) 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 \(i\).

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 \(i\).