On the L∞ convergence of a conservative Fourier pseudo‐spectral method for the space fractional nonlinear Schrödinger equation. Issue 2 (31st October 2020)

• Record Type: Journal Article
• Authors: Xu, Zhuangzhi; Cai, Wenjun; Jiang, Chaolong; Wang, Yushun
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 2(2021)
• Page Start: 1591
• Page End: 1611
• Publication Date: 2020-10-31
• Subjects: conservation laws -- discrete fractional Sobolev norm -- Fourier pseudo‐spectral method -- fractional nonlinear Schrödinger equation -- priori error estimate; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22599
• Languages: English
• ISSNs: 0749-159X

Abstract

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 ( τ 2 + N α /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.