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

Divergence-Free Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Rotating Shallow Water Magnetohydrodynamics