On the L∞ convergence of a conservative Fourier pseudo‐spectral method for the space fractional nonlinear Schrödinger equation. Issue 2 (31st October 2020) - Xu, Zhuangzhi; Cai, Wenjun; Jiang, Chaolong; Wang, Yushun
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.