On the L^∞ convergence of a conservative Fourier pseudo‐spectral method for the space fractional nonlinear Schrödinger equation

Abstract

In this paper, we take the space fractional nonlinear Schrödinger as an example to establish the L∞ convergence error analysis for the conservative Fourier pseudo‐spectral method, which has not been studied. We introduce a new fractional Sobolev norm to construct the discrete fractional Sobolev space, and also prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo‐spectral method is unconditionally convergent with order O(τ2 + Nα/2 − r) in the discrete L∞ norm, where τ is the time step and N is the number of collocation points used in the spectral method. Numerical examples are presented to verify the theoretical analysis.

Stability and Convergence Analysis of Multi-Symplectic Variational Integrator for Nonlinear Schrödinger Equation

“…Xu et al use the Fourier pseudo-spectral method to calculate the numerical solution of the space fractional nonlinear Schr ödinger equation. They prove the solution's existence and the conservation and convergence of the scheme in [25]. For a numerical solution of the Equation (1) to be discussed in this paper, a temporal two-mesh compact difference method was proposed in [12].…”