Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

TL;DR

This paper develops high-order invariant region preserving numerical schemes for relativistic hydrodynamics, ensuring solutions satisfy physical constraints by leveraging a novel explicit form of the invariant region and proving their IRP properties.

Contribution

It introduces a new explicit linear form of the invariant region for RHD, enabling the design of provably IRP high-order schemes without relying on the exact Riemann solver.

Findings

Schemes preserve physical constraints like entropy, velocity, and pressure.

First-order scheme satisfies a minimum entropy principle under CFL condition.

Numerical tests confirm the effectiveness of the proposed methods.

Abstract

This paper explores Tadmor's minimum entropy principle for the relativistic hydrodynamics (RHD) equations and incorporates this principle into the design of robust high-order discontinuous Galerkin (DG) and finite volume schemes for RHD on general meshes. The schemes are proven to preserve numerical solutions in a global invariant region constituted by all the known intrinsic constraints: minimum entropy principle, the subluminal constraint on fluid velocity, and the positivity of pressure and rest-mass density. Relativistic effects lead to some essential difficulties in the present study, which are not encountered in the non-relativistic case. Most notably, in the RHD case the specific entropy is a highly nonlinear implicit function of the conservative variables, and, moreover, there is also no explicit formula of the flux in terms of the conservative variables. In order to overcome the resulting challenges, we first propose a novel equivalent form of the invariant region, by skillfully introducing two auxiliary variables. As a notable feature, all the constraints in the novel form are explicit and linear with respect to the conservative variables. This provides a highly effective approach to theoretically analyze the invariant-region-preserving (IRP) property of schemes for RHD, without any assumption on the IRP property of the exact Riemann solver. Based on this, we prove the convexity of the invariant region and establish the generalized Lax--Friedrichs splitting properties via technical estimates, lying the foundation for our IRP analysis. It is shown that the first-order Lax--Friedrichs scheme for RHD satisfies a local minimum entropy principle and is IRP under a CFL condition. Provably IRP high-order DG and finite volume methods are developed for the RHD with the help of a simple scaling limiter. Several numerical examples demonstrate the effectiveness of the proposed schemes.