Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation

Abstract

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.