主讲人：邱建贤 教授 （厦门大学）
时 间：2026年1月5日 10:30
地 点：数学楼401报告厅
报告内容介绍：
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 efffcient 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 uniffed 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 ffnd that the unique positive root of a cubic polynomial always falls within this “safe”” interval. To enhance efffciency, 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 ffeld. 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 efffciency and robustness of the PCP NR method.
主讲人介绍：
邱建贤，厦门大学数学科学学院教授，在间断Galerkin（DG）、加权本质无振荡（WENO）数值方法的研究及应用上取得了许多重要成果，已发表学术论文一百多篇。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。