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In this paper, we introduce a new extended version of the shallow waterequations with surface tension which is skew-symmetric with respect to the L2scalar product and allows for large gradients of fluid height. This result is ageneralization of the results published by P. Noble and J.-P. Vila in [SIAM J.Num. Anal. (2016)] and by D. Bresch, F. Couderc, P. Noble and J.P. Vila in[C.R. Acad. Sciences Paris (2016)] which are restricted to quadratic forms ofthe capillary energy respectively in the one dimensional and two dimensionalsetting.This is also an improvement of the results by J. Lallement, P.Villedieu et al. published in [AIAA Aviation Forum 2018] where the augmentedversion is not skew-symetric with respect to the L2 scalar product. Based onthis new formulation, we propose a new numerical scheme and perform a nonlinearstability analysis.Various numerical simulations of the shallow water equationsare presented to show differences between quadratic (w.r.t the gradient of theheight) and general surface tension energy when high gradients of the fluidheight occur.

Augmented Skew-Symetric System for Shallow-Water System with Surface Tension Allowing Large Gradient of Density