A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations. Issue 1 (28th July 2021)

• Record Type: Journal Article
• Authors: Hu, Hongling; Jin, Xianlin; He, Dongdong; Pan, Kejia; Zhang, Qifeng
• Is Part Of: Numerical methods for partial differential equations. Volume 38:Issue 1(2022)
• Page Start: 4
• Page End: 32
• Publication Date: 2021-07-28
• Subjects: conservative difference scheme -- pointwise error estimate -- Riesz fractional derivative -- second‐order convergence -- unconditional stability; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22788
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: In this paper, a linearized semi‐implicit finite difference scheme is proposed for solving the two‐dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at the discrete level, with an unconditional stability and a second‐order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.