LAUSR

Abstract In this study, a convergent two-dimensional depth-integrated non-hydrostatic shallow water model is discretized using the discontinuous Galerkin (DG) method. This model can account for the dispersive effects by including a non-hydrostatic pressure component and can be used for the simulation of weakly dispersive water waves. Additionally, this model is based on the fractional step method. In the first step, the traditional nonlinear shallow water model plus a transport equation for vertical momentum is discretized by the DG method, and the resulting semi-discrete system is evolved in time by the explicit third-order strong-stability-preserving Runge-Kutta (SSPRK3) method to obtain the provisional solution. In the second step, the provisional solution is corrected by satisfying a divergence constraint for the velocity. The latter step is conducted after application of the DG discretization to an elliptic equation regarding the non-hydrostatic pressure. Both theoretical and experimental tests are carried out to validate the proposed model. The results indicate that for smooth problems, our model can arrive at a given error tolerance by using a mesh with fewer degrees of freedom (DOFs) at the cost of a smaller CPU time when a higher approximation order is adopted and can properly simulate wave run-up, diffraction, refraction, and focusing.