Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations

In this paper, we develop, analyze and test local discontinuous Galerkin (DG) methods to solve two classes of two-dimensional nonlinear wave equations formulated by the Kadomtsev-Petviashvili (KP) equation and the Zakharov-Kuznetsov (ZK) equation. Our proposed scheme for the Kadomtsev-Petviashvili equation satisfies the constraint from the PDE which contains a non-local operator and at the same time has the local property of the discontinuous Galerkin methods. The scheme for the Zakharov-Kuznetsov equation extends the previous work on local discontinuous Galerkin method solving one-dimensional nonlinear wave equations to the two-dimensional case. L2 stability of the schemes is proved for both of these two nonlinear equations. Numerical examples are shown to illustrate the accuracy and capability of the methods.