A Discontinuous Galerkin Method for Two-Dimensional Depth Integrated Non-hydrostatic Shallow Water Model

Abstract

A two-dimensional depth-integrated non-hydrostatic shallow water model is discretized using the quadrature-free nodal discontinuous Galerkin (NDG) method. Compared with the traditional hydrostatic shallow water model, this model includes a non-hydrostatic pressure component, which accounts for the dispersive effects ignored by the hydrostatic one, and can be used for the simulation of weakly dispersive water waves. The whole simulation strategy of the model consists of two steps. In the first step, the hydrostatic shallow water model is discretized by the quadrature-free NDG method. Then the semi-discrete system is evolved in time by a low-storage version of the fourth-order explicit Runge-Kutta method (LSERK) to obtain the intermediate solution. In the second step, the solution is corrected by satisfying a divergence constraint for the velocity. This latter step is followed by application of the DG discretization to an elliptic equation about the non-hydrostatic pressure. Tests including regular waves propagation over a submerged trapezoidal bar and 2D tsunami run-up are carried out to validate the proposed model.