Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes

Arnaud Duran (1) , Fabien Marche (1) , R. Turpault , C. Berthon

1 IMAG - Institut Montpelliérain Alexander Grothendieck

The following work is devoted to the construction and validation of a numerical scheme for the 2D shallow water system on unstructured meshes, supplemented by topography and friction source terms. Approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws, involving the water free surface and some fractions of water, accounting for the topography variations. The discretization of the friction source terms relies on the use of a modified Riemann solver for the flux computation. The resulting scheme is then corrected in order to achieve an asymptotic regime preservation. A MUSCL reconstruction is also performed to increase the space order of accuracy. The overall numerical approach is shown to be consistent, well-balanced and to satisfy a domain invariant principle. These results are assessed through several benchmark tests, involving complex geometry and varying bathymetry. In the presence of dry areas, special interest is given to the wave front speed computation, highlighting the stability of the method, even when implementing the asymptotic preserving correction.

Mots clés: Nonlinear shallow water; Asymptotic preserving; Finite volume; Friction; Well-balanced schemes; Unstructured mesh

Domaines: Analyse numérique [math.NA]

Arnaud Duran, Fabien Marche, R. Turpault, C. Berthon. Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes. Journal of Computational Physics, 2015, 287, pp.184-206. ⟨10.1016/j.jcp.2015.02.007⟩. ⟨hal-01302147⟩