Efficient and stable numerical methods for the generalized and vector Zakharov system

Abstract

We present efficient and stable numerical methods for approximations of the generalized Zakharov system (GZS) and vector Zakharov system for multicomponent plasma (VZSM) with/without a linear damping term. The key points in the methods are based on (i) a time-splitting discretization of a Schrödinger-type equation in GZS or VZSM, (ii) discretizing a nonlinear wave-type equation by a pseudospectral method for spatial derivatives, and (iii) solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The methods are explicit, unconditionally stable, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant when there is no damping term in GZS or VZSM, conserve (or keep the same decay rate of) the wave energy as that in GZS or VZSM without a (or with a linear) damping term, and give exact results for the plane-wave solution. Extensive numerical tests are presented for plane waves and solitary-wave collisions in one-dimensional GZS, and we also give the dynamics of three-dimensional VZSM to demonstrate our new efficient and accurate numerical methods. Furthermore, the methods are applied to study the convergence and quadratic convergence rates of VZSM to GZS and of GZS to the nonlinear Schrödinger (NLS) equation in the "subsonic limit" regime (0 < ε < 1), where the parameter ε is inversely proportional to the acoustic speed. Our tests also suggest that the following meshing strategy (or ε-resolution) is admissible in this regime: spatial mesh size h = O(ε) and time step k = O(ε). © 2005 Society for Industrial and Applied Mathematics.