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.