A long-standing and formidable challenge faced by all conservative schemes for relativistic magnetohydrodynamics (RMHD) is the recovery of primitive variables from conservative ones. This process involves solving highly nonlinear equations subject to physical constraints. An ideal solver should be "robust, accurate, and fast -- it is at the heart of all conservative RMHD schemes," as emphasized in [S.C. Noble et al., ApJ, 641:626-637, 2006]. Despite over three decades of research, seeking efficient solvers that can provably guarantee stability and convergence remains an open problem. This paper presents the first theoretical analysis for designing a robust, physical-constraint-preserving (PCP), and provably (quadratically) convergent Newton-Raphson (NR) method for primitive variable recovery in RMHD. Our key innovation is a unified approach for the initial guess, devised based on sophisticated analysis. It ensures that the NR iteration consistently converges and adheres to physical constraints. Given the extreme nonlinearity and complexity of the iterative function, the theoretical analysis is highly nontrivial and technical. We discover a pivotal inequality for delineating the convexity and concavity of the iterative function and establish theories to guarantee the PCP property and convergence. We also develop theories to determine a computable initial guess within a theoretical "safe" interval. Intriguingly, we find that the unique positive root of a cubic polynomial always falls within this interval. Our PCP NR method is versatile and can be seamlessly integrated into any RMHD scheme that requires the recovery of primitive variables, potentially leading to a broad impact in this field. As an application, we incorporate it into a discontinuous Galerkin method, resulting in fully PCP schemes. Several numerical experiments demonstrate the efficiency and robustness of the PCP NR method.

翻译：相对论性磁流体动力学（RMHD）所有守恒格式长期面临一个严峻挑战：从守恒变量恢复原始变量。该过程需在物理约束条件下求解高度非线性方程。正如[S.C. Noble等，ApJ, 641:626-637, 2006]所强调，理想的求解器应"稳健、精确且快速——这是所有守恒型RMHD格式的核心"。尽管经过三十余年的研究，寻找能证明保证稳定性与收敛性的高效求解器仍是一个开放性问题。本文首次提出构建RMHD原始变量恢复中鲁棒、保物理约束（PCP）且可证明（二次）收敛的牛顿-拉夫逊（NR）方法的理论分析。我们的核心创新在于基于精细分析设计初始猜想的统一方法，确保NR迭代始终收敛并满足物理约束。由于迭代函数的极端非线性和复杂性，理论分析极具挑战性且高度技术化。我们发现描述迭代函数凸凹性的关键不等式，并建立保证PCP性质与收敛性的理论。同时，我们发展出在理论"安全"区间内确定可计算初始猜想的方法。有趣的是，三次多项式唯一正根始终落在此区间内。我们的PCP NR方法具有普适性，可无缝集成至任何需要原始变量恢复的RMHD格式中，有望在该领域产生广泛影响。作为应用实例，我们将其融入间断伽辽金方法，构建了完全PCP格式。多项数值实验验证了PCP NR方法的效率与鲁棒性。