Well balanced finite volume schemes for shallow water equations on manifolds

In this paper we propose a novel second-order accurate well balanced scheme for shallow water equations in general covariant coordinates over manifolds. In our approach, once the gravitational field is defined for the specific case, one equipotential surface is detected and parametrized by a frame of general covariant coordinates. This surface is the manifold whose covariant parametrization induces a metric tensor. The model is then re-written in a hyperbolic form with a tuple of conserved variables composed both of the evolving physical quantities and the metric coefficients. This formulation allows the numerical scheme to automatically compute the curvature of the manifold as long as the physical variables are evolved. In a classical well balanced formulation, the knowledge of a given equilibrium is exploited for evolving the specific state seen as the sum of the equilibrium and the fluctuations of the state around the equilibrium itself. On the contrary, the numerical approach proposed here is automatically well balanced for the water at rest solution (defined by zero velocity and constant free surface) for general manifoldswithout having to exploit the exact equilibrium profile during the computation. In particular, this numerical strategy allows to preserve the accuracy of the water at rest equilibrium at machine precision and on large timescales, even for non-smooth bottom topographies. As a matter of fact, the proposed local polynomial reconstruction, needed at each time step by the scheme for evolving the state, is built in order to automatically cancel any numerical error committed for describing jumps in the bathymetry. As a further effect, also out of the equilibrium, the typical spurious non-physical oscillations of the recovered numerical solution in a neighborhood of the discontinuities are healed. Thus, once the information on both the bathymetry and the metric of the manifold are properly collected in the flux terms and in the nonconservative products, the resulting numerical strategy turns out to be more accurate in finding also non-equilibrium solutions. Numerical results close the work. After having proved that the scheme is second-order accurate, we test the recovery of the water at rest equilibrium at machine precision for different bottom topographies (eventually discontinuous) with various metric on very long simulation times. In particular, after having considered smooth bathymetries with compact support, we relax the assumption on the continuity of water at rest equilibrium with discontinuous bathymetries disturbed by a white noise.