The relativistic hydrodynamics (RHD) equations have three crucial intrinsic physical constraints on the primitive variables: positivity of pressure and density, and subluminal fluid velocity. However, numerical simulations can violate these constraints, leading to nonphysical results or even simulation failure. Designing genuinely physical-constraint-preserving (PCP) schemes is very difficult, as the primitive variables cannot be explicitly reformulated using conservative variables due to relativistic effects. In this paper, we propose three efficient Newton--Raphson (NR) methods for robustly recovering primitive variables from conservative variables. Importantly, we rigorously prove that these NR methods are always convergent and PCP, meaning they preserve the physical constraints throughout the NR iterations. The discovery of these robust NR methods and their PCP convergence analyses are highly nontrivial and technical. As an application, we apply the proposed NR methods to design PCP finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes incorporate high-order HWENO reconstruction, a PCP limiter, and strong-stability-preserving time discretization. We rigorously prove the PCP property of the fully discrete schemes using convex decomposition techniques. Moreover, we suggest the characteristic decomposition with rescaled eigenvectors and scale-invariant nonlinear weights to enhance the performance of the HWENO schemes in simulating large-scale RHD problems. Several demanding numerical tests are conducted to demonstrate the robustness, accuracy, and high resolution of the proposed PCP HWENO schemes and to validate the efficiency of our NR methods.

翻译：相对论流体力学（RHD）方程对原始变量有三个关键的固有物理约束：压强和密度的正定性，以及流体速度亚光速性。然而，数值模拟可能会违反这些约束，导致非物理结果甚至模拟失败。由于相对论效应，原始变量无法用守恒变量显式重构，设计真正的物理约束保持（PCP）格式极具挑战性。本文提出三种高效的牛顿-拉夫逊（NR）方法，用于从守恒变量稳健地恢复原始变量。重要的是，我们严格证明了这些NR方法始终收敛且满足PCP性质，即在NR迭代过程中始终保持物理约束。这些稳健NR方法的发现及其PCP收敛性分析具有高度非平凡性和技术性。作为应用，我们将所提出的NR方法用于设计求解RHD方程的PCP有限体积Hermite加权本质无振荡（HWENO）格式。我们的PCP HWENO格式融合了高阶HWENO重构、PCP限制器和强稳定性保持时间离散。利用凸分解技术，我们严格证明了全离散格式的PCP性质。此外，我们建议采用具有重缩放特征向量和尺度不变非线性权的特征分解，以提升HWENO格式在模拟大规模RHD问题中的性能。通过多个具有挑战性的数值测试，验证了所提PCP HWENO格式的稳健性、精度和高分辨率，并证实了我们NR方法的高效性。