We develop a new second-order flux globalization based path-conservative central-upwind (PCCU) scheme for rotating shallow water magnetohydrodynamic equations. The new scheme is designed not only to maintain the divergence-free constraint of the magnetic field at the discrete level but also to satisfy the well-balanced (WB) property by exactly preserving some physically relevant steady states of the underlying system. The locally divergence-free constraint of the magnetic field is enforced by following the method recently introduced in [A. Chertock, A. Kurganov, M. Redle, and K. Wu, ArXiv preprint (2022), arXiv:2212.02682]: we consider a Godunov-Powell modified version of the studied system, introduce additional equations by spatially differentiating the magnetic field equations, and modify the reconstruction procedures for magnetic field variables. The WB property is ensured by implementing a flux globalization approach within the PCCU scheme, leading to a method capable of preserving both still- and moving-water equilibria exactly. In addition to provably achieving both the WB and divergence-free properties, the new method is implemented on an unstaggered grid and does not require any (approximate) Riemann problem solvers. The performance of the proposed method is demonstrated in several numerical experiments that confirm the lack of spurious oscillations, robustness, and high resolution of the obtained results.

翻译：我们针对旋转浅水磁流体动力学方程组，提出了一种新型二阶通量全局化路径守恒中心迎风（PCCU）格式。该格式不仅能在离散层面维持磁场无散度约束，还能通过精确保持系统某些物理相关稳态解来满足保平衡（WB）性质。磁场的局部无散度约束通过遵循近期文献[A. Chertock, A. Kurganov, M. Redle, and K. Wu, ArXiv preprint (2022), arXiv:2212.02682]引入的方法实现：我们采用所研究系统的Godunov-Powell修正版本，通过对磁场方程进行空间微分引入附加方程，并修改磁场变量的重构过程。通过将通量全局化方法嵌入PCCU格式，确保了WB性质，从而获得了能精确保持静止与运动水流平衡状态的计算方法。新方法除可证明同时具备WB与无散度特性外，还采用非交错网格实施，且无需任何（近似）黎曼问题求解器。数值实验验证了该方法的性能，表明所得结果无虚假振荡、具有鲁棒性及高分辨率特性。