Preliminary results in Local Discontinuous Galerkin Methods for Two Classes of 2D Nonlinear Wave Equations

摘要

In this presentation, we describe our preliminary work concerning the development of the local discontinuous Galerkin method to solve two classes of two-dimensional nonlinear wave equations formulated by the Kadomtsev-Petviashvili equation and the Zakharov-Kuznetsov equation. Our proposed scheme for the Kadomtsev-Petviashvili equation is by the use of a new class of piecewise polynomial basis functions, which satisfies the constraint from the PDE containing a non-local operator and at the same time has the local property of the discontinuous Galerkin methods. The scheme is also easy for implementation and details related to implementation process are also shown. The scheme for the Zakharov-Kuznetsov equation extends the previous work on local discontinuous Galerkin method solving one-dimensional nonlinear wave equations [3, 4] to the two-dimensional setting. The methods for these two nonlinear equations can be easily designed for non-periodic boundary conditions and such non-periodic boundary conditions are used in the numerical experiments. We prove the nonlinear L stability of the schemes for both of these two nonlinear equations. We use the explicit exponential time differencing method for time discretization, which can achieve high accuracy and maintaining good stability while avoiding the severe explicit stability limit of the traditional Runge-Kutta discontinuous Galerkin methods which use explicit and nonlinearly stable high order Runge-Kutta time discretization due to the presence of the high order derivative terms. Numerical examples will be shown to illustrate the accuracy and capability of the methods.