A New Conservative Finite Difference Scheme For Klein-Gordon-Zakharov Equations

Klein-Gordon-Zakharov(KGZ)system is a system of important wave equations which describes the mutual interaction between the Langmuir waves and ion acoustic waves in a plasma.This paper aims to design and analyze numerical method for the high-dimensional KGZ equations.For the derivation of the numerical method,by introducing two auxiliary functions,we split the two second-order evolution equations in the KGZ system into four first-order evolutions.We then introduce the discrete variational method to descritize the new equivalent system to obtain a new finite difference scheme which preserves the total energy in the discrete sense.For the analysis of the numerical method,without any restriction on the grid ratios,we introduce the cut-off function technique and the lifting technique and use the standard energy method to obtain the optimal error estimates with second-order convergence both in space and in time.However,those existing results in literatures often require a certain restriction on grid ratios.For the implementation of the numerical method,we eliminate the two auxiliary functions to get an equivalent finite difference scheme which can be solved by using a technical ADI algorithm.Numerical results are reported to test the error estimates and simulate the dynamics of the KGZ system.