2016/04/18 by Arnaud Duran, Duran, Arnaud, Fabien Marche +1 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1604.05227
openalex publication_date 2016/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce a discontinuous Finite Element formulation on\nsimplicial unstructured meshes for the study of free surface flows based on the\nfully nonlinear and weakly dispersive Green-Naghdi equations. Working with a\nnew class of asymptotically equivalent equations, which have a simplified\nanalytical structure, we consider a decoupling strategy: we approximate the\nsolutions of the classical shallow water equations supplemented with a source\nterm globally accounting for the non-hydrostatic effects and we show that this\nsource term can be computed through the resolution of scalar elliptic\nsecond-order sub-problems. The assets of the proposed discrete formulation are:\n(i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary\norder of approximation in space, (iii) the exact preservation of the motionless\nsteady states, (iv) the preservation of the water height positivity, (v) a\nsimple way to enhance any numerical code based on the nonlinear shallow water\nequations. The resulting numerical model is validated through several\nbenchmarks involving nonlinear wave transformations and run-up over complex\ntopographies.\n