A conservative numerical method for the fractional nonlinear Schr(o)dinger equation in two dimensions

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（10）
• 页码范围: 1997-2014
• 页数: 18页
• 语种: 英文

摘要

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional non-linear Schr(o)dinger (NLS) equation involving fractional Laplacian.The scheme is based on a weighted and shifted Griinwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian.We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations.By introducing the differentiation matrices,the semi-discrete fractional nonlinear Schr(o)dinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrix formulations.Two kinds of time discretization methods are proposed for the semi-discrete formulation.One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation.The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort.It can be shown that the cIIF scheme can approximate CN scheme with the error O(-2).Finally numerical results are presented to demonstrate the method's conservation,accuracy,efficiency and the capability of capturing blow-up.