Low dispersion finite volume/element discretization of the enhanced Green–Naghdi equations for wave propagation, breaking and runup on unstructured meshes

Abstract

We study a hybrid approach combining a finite volume (FV) and a finite element (FE) method to solve a fully-nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard P1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in H(div) spaces. However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in H1. We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates. Finally, the full mode, coupling a wave breaking closure previously developed by the authors, is thoroughly tested on standard benchmarks using unstructured grids with sizes comparable or coarser than those usually proposed in literature.