A Flexible 2D Nonlinear Approach for Nonlinear Wave Propagation, Breaking and Run up

International audience; We present a hybrid solution strategy for the numerical solution ofthe two-dimensional (2D) partial differential equations of Green-Nagdhi(GN), which simulates fully nonlinear, weakly dispersive free surfacewaves. We re-write the standard form of the equations by splitting theoriginal system in its elliptic and hyperbolic parts, through the definitionof a new variable, accounting for the dispersive effects and having therole of a non-hydrostatic pressure gradient in the shallow water equations.We consider a two-step solution procedure. In the first step wecompute a source term by inverting the elliptic coercive operator associatedto the dispersive effects; then in a hyperbolic step we evolve theflow variables by using the non-linear shallow water equations, with allnon-hydrostatic effects accounted by the source computed in the ellipticphase. The advantages of this procedure are firstly that the GN equationsare used for propagation and shoaling, while locally reverting to the nonlinearshallow water equations to model energy dissipation in breakingregions. Secondly and from the numerical point of view, this strategyallows each step to be solved with an appropriate numerical method onarbitrary unstructured meshes. We propose a hybrid finite element (FE)finite volume (FV) scheme, where the elliptic part of the system is discretizedby means of the continuous Galerkin FE method and the hyperbolicpart is discretized using a third-order node-centred finite volume(FV) technique. The performance of the numerical model obtained is extensivelyvalidated against experimental measurements from a series ofrelevant benchmark problems.