An efficient covariant frame for the spherical shallow water equations: Well balanced DG approximation and application to tsunami and storm surge

In this work we consider an efficient discretization of the Shallow Water Equations in spherical geometry for oceanographic applications. Instead of the classical 2d-covariant or 3d-Cartesian approaches, we focus on the mixed 3d/2d form of Bernard et al. (2009) which evolves the 2d momentum tangential to the sphere by projecting the 3d-Cartesian right-hand side on the sphere surface. Differently from the last reference we consider the exact representation of the sphere instead of the finite element one, mixed with a covariant basis projection of the momentum. This leads to several simplifications of the Discontinuous Galerkin scheme: the local mass matrix goes back to the standard block-diagonal form; the Riemann Problem does not require any tensor or vector rotations to align the bases on the two sides of an edge. Second we consider well balancing corrections related to relevant equilibrium states for tsunami and storm surge simulations. These corrections allow to compensate for the inherent non-exactness of the quadrature induced by the non-polynomial nature of both the geometrical mapping and of the covariant basis. In other words, these corrections are the order of the cubature error. We show that their addition makes the scheme exactly well balanced, and is equivalent to recasting the integral of the hydrostatic pressure term in strong form. The proposed method is validated on academic benchmarks involving both smooth and discontinuous solutions, and applied to realistic tsunami and a historical storm surge simulation.