An efficient and accurate Fourier pseudo-spectral method for the nonlinear Schrödinger equation with wave operator

Abstract

In this study, we design and analyse an efficient and accurate Fourier pseudo-spectral method for solving the nonlinear Schrödinger equation with wave operator (NLSW). In this method, a modified leap-frog finite difference method is adopted for time discretization and a Fourier pseudo-spectral method is employed for spatial discretization. A new type of discrete energy functional is defined in a recursive way. It is then proved that the proposed method preserves the total energy in the discrete sense. Thanks to the FFT algorithm, the proposed method can be explicitly computed in the practical computation. Besides the standard energy method, the key tools used in our numerical analysis are a mathematical induction argument and a lifting technique. Without any restriction on the grid ratio, we prove that the proposed method is of spectral accuracy in space and second-order accuracy in time. Numerical results are reported to verify the error estimate and energy conservation of the proposed method.