High-order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State
Linfeng Xu, Shengrong Ding, Kailiang Wu
TL;DR
The paper tackles the challenge of designing high-order entropy-stable schemes for relativistic hydrodynamics with a broad, Synge-type EOS, addressing limitations of prior work restricted to the ideal EOS. It develops a complete entropy framework, including a convex entropy function, entropy variables, and an explicit two-point entropy-conservative flux, enabling high-order EC and ES schemes via ENO/WENO reconstructions and a general dissipation matrix based on the system's scaled eigenvectors. The authors extend the construction to 2D through a dimension-by-dimension approach and derive direction-specific dissipation matrices, ensuring accurate resolution of stationary contacts and robust shock capturing across multiple EOSs. A comprehensive set of 1D and 2D numerical experiments validates high-order accuracy, entropy decay in ES schemes, and the schemes’ effectiveness for four representative Synge-type EOSs, confirming practical applicability to realistic relativistic flows.
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.