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This paper explores Tadmor's minimum entropy principle for the relativistichydrodynamics (RHD) equations and incorporates this principle into the designof robust high-order discontinuous Galerkin (DG) and finite volume schemes forRHD on general meshes. The schemes are proven to preserve numerical solutionsin a global invariant region constituted by all the known intrinsicconstraints: minimum entropy principle, the subluminal constraint on fluidvelocity, and the positivity of pressure and rest-mass density. Relativisticeffects lead to some essential difficulties in the present study, which are notencountered in the non-relativistic case. Most notably, in the RHD case thespecific entropy is a highly nonlinear implicit function of the conservativevariables, and, moreover, there is also no explicit formula of the flux interms of the conservative variables. In order to overcome the resultingchallenges, we first propose a novel equivalent form of the invariant region,by skillfully introducing two auxiliary variables. As a notable feature, allthe constraints in the novel form are explicit and linear with respect to theconservative variables. This provides a highly effective approach totheoretically analyze the invariant-region-preserving (IRP) property of schemesfor RHD, without any assumption on the IRP property of the exact Riemannsolver. Based on this, we prove the convexity of the invariant region andestablish the generalized Lax--Friedrichs splitting properties via technicalestimates, lying the foundation for our IRP analysis. It is shown that thefirst-order Lax--Friedrichs scheme for RHD satisfies a local minimum entropyprinciple and is IRP under a CFL condition. Provably IRP high-order DG andfinite volume methods are developed for the RHD with the help of a simplescaling limiter. Several numerical examples demonstrate the effectiveness ofthe proposed schemes.

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