Numerical studies on the Zakharov system

Abstract

We present two numerical methods to simulate Zakharov system (ZS). The first one is the time-splitting spectral (TSSP) method, which is explicit, keeps the same decay rate of a standard variant as that in the generalized ZS, gives exact results for the plane-wave solution, and is of spectral-order accuracy in space and second-order accuracy in time. The second one is to use the discrete singular convolution (DSC) for spatial derivatives and the fourth-order Runge-Kutta (RK4) for time integration, which is of high (the same as spectral) order accuracy in space and can be applied to deal with general boundary conditions. In order to test accuracy and stability, we compare these two methods with other existing methods: Fourier pseudospectral method (FPS) and wavelet-Galerkin method (WG) for spatial derivatives combining with RK4 for time integration, as well as the standard finite difference method (FD) for solving the ZS with a solitary-wave solution. Extensive numerical tests are presented for plane waves, colliding solitary waves in 1d, a 2d problem as well as the generalized ZS with a damping term. Furthermore, extension of TSSP to standard vector ZS and ZS for multi-component plasma are presented. Numerical results show that DSC and TSSP are spectral-order accuracy in space and much more accurate than FD, and for stability, TSSP requires k=O(h), DSC-RK4 requires k=O(h^2), where k is time step and h is spatial mesh size.