# A study of discontinuous Galerkin methods for water waves simulation

## Dr. Satyendra K. Tomar

**July 14, 2005, 1 p.m. HF 136**

A discontinuous Galerkin finite element method for the numerical simulation of water waves will be presented. An accurate representation of the nonlinear waves with a significant amplitude gives rise to the deforming elements and it is well known that the standard finite element techniques with asymmetric spatial discretization lead to instabilities in the numerical solution of such problems. To overcome this problem, among the practically successful techniques are, the addition of the viscosity terms to the free surface (FS), and the finite difference (FD) reconstruction of the FS. However, both of these approaches have their own limitations, the viscosity approach unnecessarily damps the wave amplitude, and the FD approach is not attractive in general geometries and for hp-adaptation. In this talk two approaches will be discussed: (1) using the coupled form of the FS equations and implicit time discretizations, and (2) treating the FS equations in decoupled form and explicit time schemes. A detailed semi-discrete stability analysis, followed by a fully discrete stability analysis of RK4 scheme of the second approach will be presented. Further, to improve the accuracy of the computed velocity field a superconvergent gradient recovery technique based on L2 projections has been employed and the supporting results from the stability analysis will be presented.