Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine–Gordon equation with damping

Abstract

In this paper, an efficient numerical scheme is presented for solving the space fractional nonlinear damped sine–Gordon equation with periodic boundary condition. To obtain the fully-discrete scheme, the modified Crank–Nicolson scheme is considered in temporal direction, and Fourier pseudo-spectral method is used to discretize the spatial variable. Then the dissipative properties and spectral-accuracy convergence of the proposed scheme in L∞ norm in one-dimensional (1D) space are derived. In order to effectively solve the nonlinear system, a linearized iteration based on the fast Fourier transform algorithm is constructed. The resulting algorithm is computationally efficient in long-time computations due to the fact that it does not involve matrix inversion. Extensive numerical comparisons of one- and two-dimensional (2D) cases are reported to verify the effectiveness of the proposed algorithm and the correctness of the theoretical analysis.