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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:419

Article

Bresch, D.¹  Cellier, N.²  Couderc, F.³  Gisclon, M.¹  Noble, P.³  Richard, G-L¹  Ruyer-Quil, C.²  Vila, J-P³ 

[1] Univ Savoie Mt Blanc, LAMA UMR5127 CNRS, F-73376 Le Bourget Du Lac, France

[2] Univ Savoie Mt Blanc, LOCIE UMR5271 CNRS, F-73376 Le Bourget Du Lac, France

[3] INSA Toulouse, Inst Math Toulouse, UMR5219 CNRS, F-31077 Toulouse 4, France

关键词: Shallow-water;    Surface tension;    Augmented system;    Numerical scheme;    Non-linear stability;    Numerical simulations;   

DOI  :  10.1016/j.jcp.2020.109670

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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