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

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