Conservative Fourier spectral method and numerical investigation of space fractional Klein–Gordon–Schrödinger equations

Abstract

In this paper, we propose Fourier spectral method to solve space fractional Klein–Gordon–Schrödinger equations with periodic boundary condition. First, the semi-discrete scheme is given by using Fourier spectral method in spatial direction, and conservativeness and convergence of the semi-discrete scheme are discussed. Second, the fully discrete scheme is obtained based on Crank–Nicolson/leap-frog methods in time direction. It is shown that the scheme can be decoupled, and preserves mass and energy conservation laws. It is proven that the scheme is of the accuracy O(τ2+N−r). Last, based on the numerical experiments, the correctness of theoretical results is verified, and the effects of the fractional orders α, β on the solitary solution behaviors are investigated. In particular, some interesting phenomena including the quantum subdiffusion are observed, and complex dynamical behaviors are shown clearly by many intuitionistic images.