An efficient 3d/2d-covariant formulation of the spherical shallow water equations: well balanced DG approximation and application to tsunami and storm surge
Résumé
In this work we consider an efficient discretization of the Shallow Water Equations in spherical geometry for oceanographic applications. Instead of classical 2d-covariant or 3d-Cartesian approaches, we focus on the mixed 3d/2d form of [Bernard et al., JCP 2009] which evolves the 2d momentum tangential to the sphere by projecting the 3d-Cartesian right-hand side on the sphere surface. First, by considering a covariant representation of the sphere instead of the finite element one, we show a simplification of the Discontinuous Galerkin scheme: local mass matrix goes back to the standard block-diagonal form, Riemann Problem do not imply tensor/vector rotation. Second we consider well-balancing corrections related to relevant equilibrium states for tsunami and storm surge simulations. These corrections are zero for the exact solution, and otherwise of the order of the quadrature formulas used. We show that their addition to the scheme is equivalent to resorting to the strong form the integral of the hydrostatic pressure term. The method proposed is validated on academic benchmarks involving both smooth and discontinuous solutions, and applied to realistic tsunami and an historical storm surge simulation.
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