Chi-Wang Shu
Division of Applied Mathematics,
Brown University
In this talk we will first briefly describe the discontinuous Galerkin methods, then introduce our recent research on developing local discontinuous Galerkin methods for solving various nonlinear dispersive wave equations. These equations include the classical KdV equations and generalized KdV equations, the K(m,n) equations with "compacton" solutions, KdV-Burgers type equations, general fifth-order KdV type equations, the fully nonlinear K(n,n,n) equations, Kuramoto-Sivashinsky equations, Ito-type coupled KdV equations, two-dimensional Kadomtsev-Petviashvili equation and Kadomtsev-Petviashvili equation, etc. For all these schemes, the local discontinuous Galerkin methods are designed to maintain provable cell entropy inequality and L^2 stability for the nonlinear cases. The methods are easy to implement on parallel machines and have their flexibility on h-p adaptivity. Numerical results will be shown to demonstrate the good behavior of these numerical schemes. The results in this talk are from joint works with J. Yan, D. Levy and J. Yan, and with Y. Xu.