Physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state

TL;DR

This paper introduces high-order, physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state, ensuring physical admissibility and high accuracy in complex astrophysical flows.

Contribution

It develops novel high-order DG methods with PCP limiting for RHD equations using a general EOS, ensuring positivity, stability, and accuracy.

Findings

Methods preserve positivity of density, pressure, and energy.

Numerical examples demonstrate robustness with large Lorentz factors and discontinuities.

High-order accuracy maintained in complex relativistic flows.

Abstract

The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics. The paper develops high-order accurate physical-constraints-preserving (PCP) central discontinuous Galerkin (DG) methods for the one- and two-dimensional special relativistic hydrodynamic (RHD) equations with a general EOS. It is built on the theoretical analysis of the admissible states for the RHD and the PCP limiting procedure enforcing the admissibility of central DG solutions. The convexity, scaling and orthogonal invariance, and Lax-Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form, and then the high-order central DG methods with the PCP limiting procedure and strong stability preserving time discretization are proved to preserve the positivity of the density, pressure, and specific internal energy, and the bound of the fluid velocity, maintain the high-order accuracy, and be $L^1$-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density or pressure etc.