报告题目：Provably Convergent Newton–Raphson Method: Theoretically Robust Recovery of Primitive Variables in Relativistic MHD

报告时间：2025-09-04   10:00-11:00

报  告 人 ：邱建贤  教授（厦门大学）

报告地点：雷军科技楼一楼报告厅（B102）

Abstract: 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 prov- ably guarantee stability and convergence remains an open problem. This paper presents the first theoretical analysis for designing a robust, physical-constraint-preserving (PCP), and prov- ably (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.