Energy-Conservative Finite Difference Method for the Coupled Nonlinear Klein-Gordon Equation in the Nonrelativistic Limit Regime

Abstract

In this paper, we propose an energy-conservative finite difference time domain (FDTD) method for solving the coupled nonlinear Klein-Gordon equations (CNKGEs) in the nonrelativistic limit regime, involving a small parameter $0 < ε ≪ 1$ which is inversely proportional to the speed of light. Employing cut-off technique, we analyze rigorously error estimates for the numerical method. Numerical results are reported to confirm the energy-conservative property and the error results in $l^2$ norm and $H^1$ norm under different values of $ε.$