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

Abstract

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 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.