Augmented skew-symmetric system for shallow-water system with surface tension allowing large gradient of density

Abstract

In this paper, we introduce a new extended version of the shallow-water equations with surface tension which may be decomposed into a hyperbolic part and a second order derivative part which is skew-symmetric with respect to the L2 scalar product. This reformulation allows for large gradients of fluid height simulations using a splitting method. This result is a generalization of the results published by Noble and Vila (2016) [24] and by Bresch et al. (2016) [3] which are restricted to quadratic forms of the capillary energy respectively in the one dimensional and two dimensional setting. This is also an improvement of the results by J. Lallement, P. Villedieu et al. published in Lallement et al. (2018) [22] where the augmented version is not skew-symmetric with respect to the L2 scalar product. Based on this new formulation, we propose a new numerical scheme and perform a nonlinear stability analysis. Various numerical simulations of the shallow water equations are presented to show differences between quadratic (w.r.t. the gradient of the height) and general surface tension energy when high gradients of the fluid height occur.