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 $L^2$ 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 $L^2$ 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.

在本文中，我们介绍了具有表面张力的浅水方程的新扩展版本，该方程可以分解为双曲部分和相对于偏斜对称的二阶导数部分。 ${\mathrm{大号}}^2$标量积。这种重新设计允许使用拆分方法对流体高度模拟进行较大的梯度转换。该结果是Noble和Vila（2016）[24]和Bresch等人发表的结果的概括。（2016）[3]分别限制在一维和二维环境下毛细管能的二次形式。这也是J. Lallement，P. Villedieu等人的结果的改进。发表于Lallement等。（2018）[22]，其中增强版本相对于${\mathrm{大号}}^2$标量积。基于此新公式，我们提出了一种新的数值方案并进行了非线性稳定性分析。给出了浅水方程的各种数值模拟，以显示当流体高度出现高梯度时，二次方（扭曲了高度的梯度）与一般表面张力能量之间的差异。