High-order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State

TL;DR

This paper introduces high-order entropy stable finite difference schemes for relativistic hydrodynamics with general Synge-type equations of state, overcoming limitations of previous ideal EOS-based methods and ensuring physical consistency.

Contribution

It develops a unified framework for entropy stable schemes applicable to a wide range of Synge-type EOSs in relativistic hydrodynamics, including novel flux constructions and dissipation matrices.

Findings

Schemes achieve high-order accuracy and entropy stability.

Numerical tests validate effectiveness for various EOSs.

Accurately resolve stationary contact discontinuities.

Abstract

All the existing entropy stable (ES) schemes for relativistic hydrodynamics (RHD) in the literature were restricted to the ideal equation of state (EOS), which however is often a poor approximation for most relativistic flows due to its inconsistency with the relativistic kinetic theory. This paper develops high-order ES finite difference schemes for RHD with general Synge-type EOS, which encompasses a range of special EOSs. We first establish an entropy pair for the RHD equations with general Synge-type EOS in any space dimensions. We rigorously prove that the found entropy function is strictly convex and derive the associated entropy variables, laying the foundation for designing entropy conservative (EC) and ES schemes. Due to relativistic effects, one cannot explicitly express primitive variables, fluxes, and entropy variables in terms of conservative variables. Consequently, this highly complicates the analysis of the entropy structure of the RHD equations, the investigation of entropy convexity, and the construction of EC numerical fluxes. By using a suitable set of parameter variables, we construct novel two-point EC fluxes in a unified form for general Synge-type EOS. We obtain high-order EC schemes through linear combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES schemes are achieved by incorporating dissipation terms into the EC schemes, based on (weighted) essentially non-oscillatory reconstructions. Additionally, we derive the general dissipation matrix for general Synge-type EOS based on the scaled eigenvectors of the RHD system. We also define a suitable average of the dissipation matrix at the cell interfaces to ensure that the resulting ES schemes can resolve stationary contact discontinuities accurately. Several numerical examples are provided to validate the accuracy and effectiveness of our schemes for RHD with four special EOSs.