相对论欧拉方程的高阶全离散多熵稳定且保界格式
High-order fully discrete multi-entropy-stable and bound-preserving schemes for relativistic Euler equations
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中文总结 AI 辅助
本文针对相对论欧拉方程，首次构造全离散多熵稳定且保界的高阶格式，通过单元投影实现，在多类算例中验证了其精度与稳定性。
中文摘要 AI 辅助
离散熵不等式是守恒律系统可用的主要非线性稳定性估计，其评估以物理上可容许的状态为前提，但迄今为止，这两者是分别被保障的。熵稳定格式几乎总是半离散的，围绕选定的单个熵对构建，且默认密度和压力为正（这是熵有定义的首要条件）；而保界限流器可保持解的可容许性，但不提供熵估计。对于特殊相对论欧拉方程，两者完全无法分离，因为保守量到原始变量的映射是隐式的，不可容许状态因此没有熵可供修正。本文中，我们构造了高阶间断伽辽金和有限体积格式，据我们所知，这是首次针对任意给定的有限凸熵对族，实现全离散意义下的熵稳定（该特性我们称为多熵稳定性），且在评估熵的所有位置均保证解可容许。所有这些都通过单个单元投影实现，既不损失守恒性也不损失设计阶数。该构造基于相对论因果性，其将所有特征速度限制为光速，因此数值粘性可针对所有状态和所有状态方程固定，且一个两点基本构件即可服务整个熵族。由于仅使用了可容许集的凸性和该速度限制，相同方法也可推广到相关系统。最后，在采用四个状态方程的计算中，该格式在接近真空时仍保持高阶精度，在强激波、近真空激波-涡旋相互作用或洛伦兹因子超过70的喷流中不产生不可容许状态，且证实了所有强制离散熵的单调衰减。
英文摘要
A discrete entropy inequality is the principal nonlinear stability estimate available for systems of conservation laws, and evaluating it presupposes a physically admissible state. So far, however, the two have been secured separately. Entropy-stable schemes are almost always semi-discrete, are built around one selected entropy pair, and take for granted the positivity of density and pressure that makes the entropy well defined in the first place, whereas bound-preserving limiters keep the solution admissible but deliver no entropy estimate. For the special relativistic Euler equations, the two cannot be separated at all, since the conservative-to-primitive map is implicit, and an inadmissible state therefore has no entropy to correct. Here we construct high-order discontinuous Galerkin and finite volume schemes that, to our knowledge, for the first time, are entropy stable in the fully discrete sense for an arbitrary prescribed finite family of convex entropy pairs, a property we call multi-entropy stability, and are provably admissible wherever an entropy is evaluated. All of this is achieved by a single cellwise projection, and neither conservation nor the design order is lost. The construction rests on relativistic causality, which bounds every characteristic speed by the speed of light. Consequently, the numerical viscosity can be fixed once for all states and all equations of state, and one two-point building block then serves the whole entropy family. Since only the convexity of the admissible set and this speed bound are used, the same route remains open for related systems. Finally, in computations with four equations of state, the schemes retain high-order accuracy close to vacuum, produce no inadmissible state in strong shocks, near-vacuum shock--vortex interaction or jets with Lorentz factor above $70$, and confirm the monotone decay of every enforced discrete entropy.
