Séance Séminaire

Séminaire ACSIOM

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction

This work deals with the numerical approximation of solutions to the shallow-water model equipped with the topography and nonlinear Manning friction source terms. In geophysical applications, steady state solutions of this model are particularly relevant, and exactly capturing these steady solutions is essential to ensure the accuracy of the numerical method for important regimes. Therefore, we seek to impose that the scheme exactly preserves the steady solutions with nonzero velocity : such a scheme is called fully well-balanced. To address this issue, we elect to derive a 1D numerical scheme within the framework of Godunov-type finite volume schemes, based on approximate solutions to Riemann problems. We design a relevant approximate Riemann solver which allows the scheme to be fully well-balanced, to preserve the non-negativity of the water height, and to exhibit a discrete entropy inequality. We extend this scheme to two space dimensions, making sure that the properties of the 1D scheme are still verified by the 2D scheme. We then propose a high-order extension of the 2D scheme. This high-order extension uses both a MOOD method and a convex combination in order to recover the properties of the first-order scheme. Numerical results, including a large-scale geophysical simulation, are finally presented to highlight the properties of the scheme.