# Solving the Riemann Problem for Realistic Astrophysical Fluids

## TL;DR

This paper introduces new methods for solving the Riemann problem with realistic equations of state in astrophysical fluids, enabling more accurate and efficient modeling of complex astrophysical flows.

## Contribution

It develops both exact and approximate Riemann solvers for general EoS, guaranteeing solution existence and uniqueness under monotonicity, and compares two thermodynamic calculation methods.

## Key findings

- Exact and approximate Riemann solutions match for perfect gas EoS.
- Analytic EoS solver is faster than interpolation-based methods.
- Numerical results show similar accuracy for both EoS solving methods.

## Abstract

We present new methods to solve the Riemann problem both exactly and approximately for general equations of state (EoS) to facilitate realistic modeling and understanding of astrophysical flows. The existence and uniqueness of the new exact general EoS Riemann solution can be guaranteed if the EoS is monotone regardless of the physical validity of the EoS. We confirm that: (1) the solution of the new exact general EoS Riemann solver and the solution of the original exact Riemann solver match when calculating perfect gas Euler equations; (2) the solution of the new Harten-Lax-van Leer-Contact (HLLC) general EoS Riemann solver and the solution of the original HLLC Riemann solver match when working with perfect gas EoS; and (3) the solution of the new HLLC general EoS Riemann solver approaches the new exact solution. We solve the EoS with two methods, one is to interpolate 2D EoS tables by the bi-linear interpolation method, and the other is to analytically calculate thermodynamic variables at run-time. The interpolation method is more general as it can work with other monotone and realistic EoS while the analytic EoS solver introduced here works with a relatively idealized EoS. Numerical results confirm that the accuracy of the two EoS solvers is similar. We study the efficiency of these two methods with the HLLC general EoS Riemann solver and find that analytic EoS solver is faster in the test problems. However, we point out that a combination of the two EoS solvers may become favorable in some specific problems. Throughout this research, we assume local thermal equilibrium.