  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.
