A long-standing and formidable challenge faced by all conservative numerical schemes for relativistic magnetohydrodynamics (RMHD) equations 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., Astrophys. J., 641 (2006), pp. 626-637]. 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, carefully devised based on sophisticated analysis. It ensures that the resulting NR iteration consistently converges and adheres to physical constraints through- out all NR iterations. 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 general auxiliary 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 "safe" interval. To enhance efficiency, we propose a hybrid strategy that combines this with a more cost-effective initial value. The presented PCP NR method is versatile and can be seamlessly integrated into any RMHD numerical scheme that requires the recovery of primitive variables, potentially leading to a very broad impact in this field. As an application, we incorporate it into a discontinuous Galerkin method, resulting in fully PCP schemes. Several numerical experiments, including random tests and simulations of ultra-relativistic jet and blast problems, demonstrate the notable efficiency and robustness of the PCP NR method.

邱建贤, 厦门大学数学科学学院教授, 国际著名刊物 Journal of Computational Physics (计算物理)编委。从事计算流体力学及微分方程数值解法的研究工作, 在间断Galerkin(DG)、加权本质无振荡(WENO)数值方法的研究及其应用方面取得了一些重要成果, 目前已经发表论文一百多篇。主持国家自然科学基金重点项目、联合基金重点支持项目和国家重点研发项目子课题各一项, 参与欧盟第六框架特别研究项目, 是项目组中唯一非欧盟的成员, 多次应邀在国际会议上作大会报告。获2020年度教育部自然科学二等奖, 2021年度福建省自然科学奖二等奖各一项。