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.
翻译:本文探讨塔德莫尔对相对性流体动力学(RHD)方程式的最低精度原则,并将这一原则纳入强力高分不连续的Galerkin(DG)和RHD一般中间值的有限体积计划的设计中。 事实证明,在由所有已知内在限制因素构成的全球异变区域,这些办法可以保留数字解决方案: 最小增温原则, 对流体速度的次光限制, 以及压力和休息质量密度的假设性。 相对性效应导致本研究中的一些基本困难, 而在非相对性估计中并未遇到这些困难。 最值得注意的是, 在RHD(DG) 中, 特定的精度是保守性不线性不隐含的保守变量功能, 此外, 也没有明确的公式, 为了克服由此产生的挑战, 我们首先提出一种新式变异性区域, 精度提出了两种辅助变量。 一个显著的特征是, 新式形式的帮助形式中的所有限制, 与保守的RRV(RRF) 规则的精度规则的精度的精度值基础, 提供了一种高度的理论分析。