标题： 具有物理约束保持性质的相对论流体力学中Hermit WENO格式的Newton-Raphson方法的收敛性分析及原始变量恢复的应用 显示英文标题

标题： Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

摘要： 相对论流体力学（RHD）方程对原始变量有三个重要的内在物理约束：压力和密度的正值性以及亚光速流体速度。然而，数值模拟可能会违反这些约束，导致非物理的结果甚至模拟失败。设计真正保持物理约束守恒（PCP）的格式非常困难，因为由于相对论效应，原始变量不能用守恒变量显式重写。本文提出了三种高效的牛顿-拉弗森（NR）方法，用于从守恒变量中稳健地恢复原始变量。重要的是，我们严格证明了这些NR方法总是收敛且保持PCP性质，这意味着它们在整个NR迭代过程中保持物理约束。发现这些稳健的NR方法及其PCP收敛分析具有高度的技术性和非平凡性。作为一个应用，我们将所提出的NR方法应用于设计求解RHD方程的PCP有限体积Hermite加权本质上非振荡（HWENO）方案。我们的PCP HWENO方案结合了高阶HWENO重构、PCP限制器和强稳定性保持的时间离散化。我们使用凸分解技术严格证明了全离散格式的PCP性质。此外，我们建议使用重新标度特征向量和尺度不变非线性权重来增强HWENO格式在模拟大规模RHD问题中的性能。进行了几个具有挑战性的数值测试，以展示所提出的PCP HWENO格式的鲁棒性、精度和高分辨率，并验证我们的NR方法的效率。 显示英文摘要

摘要： 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.