Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations

In this paper we develop a local discontinuous Galerkin method to solve the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. The L2 stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high order Runge–Kutta methods and the explicit exponential time differencing method for the time discretization; the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher order spatial derivatives. Numerical examples are shown to demonstrate the accuracy and capability of these methods.