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

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

\[ \frac{\partial}{\partial t}\left(\rho v_{i}\right) + \frac{1}{\sqrt{\gamma}} \frac{\partial}{\partial x^{j}} \left( \sqrt{\gamma} \left(\rho v_{i} v^{j} + p \delta^{j}_{\, i}\right) \right) = \frac{1}{2}\left(v^{j} v^{k} + p \gamma^{jk}\right) \frac{\partial}{\partial x^{i}}\left(\gamma_{jk}\right) - \rho \frac{\partial \Phi}{\partial x^{i}},\]

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

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

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

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 \(Y_{e} \neq 0\), 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

\[ \frac{\partial \bf{U}}{\partial t} + \frac{1}{\sqrt{\gamma}} \frac{\partial}{\partial x^{j}} \left( \sqrt{\gamma} \bf{F}^{j}(\bf{U}) \right) = \bf{S}(\bf{U}),\]

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