Physical Overview

This solver solves the Euler equations of non-relativistic hydrodynamics. These equations represent the conservation of mass, momentum, and energy in a non-viscous fluid. In a curvilinear coordinate system with metric tensor components ${\gamma }_{ij}$, they are written as

$$\frac{\partial \rho }{\partial t}+\frac{1}{\sqrt{\gamma }}\frac{\partial }{\partial x^j}\left(\sqrt{\gamma }\rho v^j\right)=0,$$

$$\frac{\partial }{\partial t}\left(\rho v_i\right)+\frac{1}{\sqrt{\gamma }}\frac{\partial }{\partial x^j}\left(\sqrt{\gamma }(\rho v_i{v}^j+p{\delta }_i^j)\right)=\frac{1}{2}(v^j{v}^k+p{\gamma }^{jk})\frac{\partial }{\partial x^i}\left({\gamma }_{jk}\right)-\rho \frac{\partial \mathrm{\Phi }}{\partial x^i},$$

$$\frac{\partial E}{\partial t}+\frac{1}{\sqrt{\gamma }}\frac{\partial }{\partial x^j}\left(\sqrt{\gamma }(E+p)v^j\right)=-\rho v^j\frac{\partial \mathrm{\Phi }}{\partial x^j},$$

where $\rho$ is the mass density, $v^i$ is the i-th component of the velocity, $p$ is the pressure, $E=\rho ϵ+\frac{1}{2}\rho v^i{v}_i$ is the energy density with specific internal energy $ϵ$, $\mathrm{\Phi }$ is the gravitational potential, and $\sqrt{\gamma }$ is the determinant of the metric. This system is closed by both Poisson’s equation

$$\frac{1}{\sqrt{\gamma }}\frac{\partial }{\partial x^i}\left(\sqrt{\gamma }{\gamma }^{ij}\frac{\partial \mathrm{\Phi }}{\partial x^j}\right)=4\pi G\rho ,$$

with the gravitational constant $G$, and a general equation of state of the form $p=p(\rho ,T,Y_e)$, where $T$ is the temperature and $Y_e$ is the electron fraction. If $Y_e\ne 0$, then we take $\rho$ to be the baryon mass density, and evolve the electron mass density ${\rho }_e=\rho Y_e$ separately with the equation

$$\frac{\partial {\rho }_e}{\partial t}+\frac{1}{\sqrt{\gamma }}\frac{\partial }{\partial x^j}\left(\sqrt{\gamma }{\rho }_e{v}^j\right)=0.$$

The conservation equations can then be combined into the single equation

$$\frac{\partial \mathbf{U}}{\partial t}+\frac{1}{\sqrt{\gamma }}\frac{\partial }{\partial x^j}(\sqrt{\gamma }{\mathbf{F}}^{\mathbf{j}}\mathbf{(}\mathbf{U}\mathbf{)})=\mathbf{S}\mathbf{(}\mathbf{U}\mathbf{)}\mathbf{,}$$

where $\mathbf{U}$ is the vector of conserved variables, ${\mathbf{F}}^{\mathbf{j}}\mathbf{(}\mathbf{U}\mathbf{)}$ is the flux vector in the j-th direction, and $\mathbf{S}\mathbf{(}\mathbf{U}\mathbf{,}\mathbf{\Phi }\mathbf{)}$ is the source vector.