The energy-preserving finite difference methods and their analyses for system of nonlinear wave equations in two dimensions

Abstract

The coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave equations. However, the energy preserving finite difference methods (EP-FDMs) for one-dimensional single sine-Gordon equation and one-dimensional single Klein-Gordon equation, can not directly be generalized to solve the systems of coupled SG or coupled KG equations, and the theoretical analysis technique used for 1D single SG equation or for 1D single KG equation is not suitable for the analysis of high dimensional problems. In this paper, we develop and analyze two kinds of energy preserving FDMs for the systems of coupled SG equations or coupled KG equations in two dimensions. One proposed scheme is a two-level scheme and the other is a three-level scheme. We prove the schemes to satisfy the energy conservations in the discrete forms. By using the fixed point theorem, it is shown that they are solvable. Also, it is further proved that they have the second order convergence rate in both time and space steps. Numerical tests show the performance of the methods and confirm the theoretical findings.