Explicit Relaxation Technique for Solving the Green-Naghdi Equations for Dispersive Waves

We introduce a relaxation technique for solving the Green-Naghdi equations for dispersive waves. We propose a numerical method that is explicit in time and uses continuous finite elements for the approximation in space. The numerical method is compatible with dry states and is provably positivity preserving under a CFL condition. The method is then numerically validated against manufactured solutions and is illustrated by comparison with laboratory experiments. We highlight some coastal engineering applications of the model such as tsunami and storm surge waves propagation over complex topographies. We show the robustness and reproducibility of the method by also implementing it in the open source software Proteus, which is developed at the U.S. Army Engineer Research and Development Center.