A Hybrid Discontinuous/Continuous Galerkin Approach to Non-Hydrostatic Euler Model for Water Waves

Coastal communities and ecosystems in near-shore areas are prone to the impacts of hurricanes, winter storms, tsunamis, and chronic sea-level rise. Numerical modeling has become an indispensable tool in coastal engineering and science. Rapid advances in computational technology allow solving the Euler equations or Navier-Stokes equations for non-hydrostatic free surface waves directly for practical applications. This work presents a non-hydrostatic Euler model for fully dispersive nonlinear water waves, and its application to wave propagation from deep sea to the shoreline. The σ-coordinate maps the irregular physical domain to the regular computation domain. A numerical scheme based on a hybrid of discontinuous Galerkin (DG) method and continuous Galerkin (CG) method is proposed. This scheme retains the advantages of Galerkin methods such as high-order approximation, hp-adaptivity (element size and polynomial order, respectively), easy handling complex geometry, etc., and reduces the computational cost by coupling the CG and DG method.

The numerical algorithm is split into hydrostatic and non-hydrostatic phases. In the hydrostatic phase, the DG method is employed to compute the mass and momentum flux explicitly. In the following non-hydrostatic phase, the CG method is adopted to solve the Poisson equation to obtain pressure correction. Model validation is carried out by conducting numerical test cases of wave propagation over a submerged bar, wave shoaling, breaking and run-up. The numerical results are compared with analytical solutions and experimental data. Detailed results will be presented at the conference.