A well-balanced finite volume solver for the 2D shallow water magnetohydrodynamic equations with topography

In this paper, a second-order finite volume Non-Homogeneous Riemann Solver is used to obtain an approximate solution for the two-dimensional shallow water magnetohydrodynamic (SWMHD) equations considering non-flat bottom topography. We investigate the stability of a perturbed steady state, as well as the stability of energy in these equations after a perturbation of a steady state using a dispersive analysis. To address the elliptic constraint $\nabla \cdot hB=0$, the GLM (Generalized Lagrange Multiplier) method designed specifically for finite volume schemes, is used. The proposed solver is implemented on unstructured meshes and verifies the exact conservation property. Several numerical results are presented to validate the high accuracy of our schemes, the well-balanced, and the ability to resolve smooth and discontinuous solutions. The developed finite volume Non-Homogeneous Riemann Solver and the GLM method offer a reliable approach for solving the SWMHD equations, preserving numerical and physical equilibrium, and ensuring stability in the presence of perturbations.