Well-balanced discontinuous Galerkin method with flux globalization for rotating shallow water equations

In this paper, we introduce a novel well-balanced discontinuous Galerkin (DG) method for the rotating shallow water equations, which is founded on the flux globalization approach. Our method entails the integration of the source term into the global fluxes, thereby establishing a quasi-conservative formulation of the equations. The well-balanced property is maintained by ensuring the equilibrium at the nodes of the Lagrange DG basis and through a tailored treatment of the numerical flux. Furthermore, we employ linear segment paths between equilibrium variables at the cell interface to preserve the equilibrium state of the scheme. This strategy allows us to handle more complex equilibrium states, including those with spatially global integral quantities, and accommodates discontinuous bottom topography. We conduct a comprehensive series of numerical experiments on shallow water models, including those with and without Coriolis forces. These experiments confirm the high-order accuracy of our DG method and its ability to exactly preserve equilibrium for intricate moving steady states. Additionally, the method successfully propagates small perturbations of the steady state with high-resolution and oscillation-free solutions, even in the presence of challenging bottom topography conditions.