A local energy-preserving scheme for Klein–Gordon–Schrödinger equations**Project supported by the National Natural Science Foundation of China (Grant Nos. 11201169, 11271195, and 41231173) and the Project of Graduate Education Innovation of Jiangsu Province, China (Grant No. CXLX13 366).

Abstract

A local energy conservation law is proposed for the Klein–Gordon–Schrödinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving (LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order \U0001d4aa(τ2 + h2). The theoretical properties are verified by numerical experiments.