An explicit residual based approach for shallow water flows

We describe a fully explicit residual based construction to discretize the shallow water equations with friction on unstructured grids. The approach is by construction exactly well balanced for al the simple known steady equilibria, and it shows a super-covergent behavior for smooth non-trivial equilibria, as the implicit residual schemes considered in (Ricchiuto, J.Sci.Comp. 48,2011). Moreover, by adapting the wetting drying technique discussed in (Ricchiuto and Bolermann, JCP 228, 2009) the scheme is able to preserve the non-negativity of the depth and to handle very well the runup and flooding of complex bathymetries. The approach is analyzed and tested thoroughly. The numerical results on a large variety of problems show the interest in the approach. Key-words: free surface flow, shallow water equations, residual approach, residual distribution, explicit scheme, rung kutta scheme, unstructured grids, second order of accuracy, C-property, generalized C-property, positivity preservation, wetting-drying ∗ INRIA Bordeaux Sud-Ouest, 200 avenue de la Vieille Tour, 33405 Talence Cedex, France [email protected] ha l-0 08 55 64 5, v er si on 1 29 A ug 2 01 3 An explicit residual based approach for shallow water flows Résumé : On considère une construction explicite et residual based pour la discrétisation des équations shallow water sur maillages non-structurés. L’approche est par construction exactement well balanced pour tous les équilibres plus simples connus en littérature, mais il montre une super-convergence pour certains équilibres non-triviaux, comme les schémas implicites considérés en (Ricchiuto, J.Sci.Comp. 48,2011). En adaptant la stratégie de wetting/drying prsentée en (Ricchiuto and Bolermann, JCP 228, 2009) le schéma permet en plus de garantir la non-ngativité de la profondeur et de simuler runup et inondation de bathymétries complexes. La méthode est analysée et testée sur un nombre important de problèmes. Les résultats montrent l’intéret de l’approche Mots-clés : free surface flow, shallow water equations, residual approach, residual distribution, explicit scheme, rung kutta scheme, unstructured grids, second order of accuracy, C-property, generalized C-property, positivity preservation, wetting-drying ha l-0 08 55 64 5, v er si on 1 29 A ug 2 01 3 Explicit residual based shallow water 3