A well-balanced, positive, entropy-stable, and multi-dimensional-aware finite volume scheme for 2D shallow-water equations with unstructured grids

Alessia Del Grosso (1, 2) , Manuel J. Castro (3) , Agnes Chan (1, 2) , Gérard Gallice (4) , Raphaël Loubère (2) , Pierre-Henri Maire (1)

1 CESTA - Centre d'études des systèmes et des technologies avancées
2 IMB - Institut de Mathématiques de Bordeaux
3 Universidad de Málaga [Málaga] = University of Málaga [Málaga]
4 Retired

In this article, we present a multi-dimensional-aware Eulerian Riemann solver (RS) and its associated finite volume (FV) scheme for the 2D shallow-water equations. This RS, appropriately derived from its associated Lagrangian version, presents the specific feature of coupling all cells in the vicinity of the current one. Consequently, this solver is no longer a 1D RS across one edge. Contrarily, it encounters for genuine multidimensional effects and for the presence of the source term of the SW equations. The associated first order FV numerical scheme ensures well-balanced for lake at rest steady states, positivity preservation and entropy stability properties. Moreover, a second-order accurate extension is proposed based on Runge-Kutta time discretization and piecewise linear limited reconstructions, that preserve the well-balanced character of the first order scheme. We present several 2D tests assessing the good behaviors of the obtained numerical scheme on unstructured mesh. The numerical scheme seems insensitive to spurious numerical instabilities such as the carbuncle effect.

Mots clés: Finite volume schemes; Lagrangian Riemann solver; Eulerian Riemann solver; Shallow-water equations; Balance laws; Well-balanced scheme

Domaines: Mathématiques [math]

Alessia Del Grosso, Manuel J. Castro, Agnes Chan, Gérard Gallice, Raphaël Loubère, et al.. A well-balanced, positive, entropy-stable, and multi-dimensional-aware finite volume scheme for 2D shallow-water equations with unstructured grids. 2023. ⟨hal-04177987⟩