A local refinement purely meshless scheme for time fractional nonlinear Schrodinger equation in irregular geometry region  

作　　者：Tao Jiang Rong-Rong Jiang Jin-Jing Huang Jiu Ding Jin-Lian Ren 蒋涛;蒋戎戎;黄金晶;丁玖;任金莲(School of Mathematical Sciences,Yangzhou University,Yangzhou 225002,China;Department of Mathematics,University of Southern Mississippi,Hattiesburg,MS 39406-5045,USA)

机构地区：[1]School of Mathematical Sciences,Yangzhou University,Yangzhou 225002,China [2]Department of Mathematics,University of Southern Mississippi,Hattiesburg,MS 39406-5045,USA

出　　处：《Chinese Physics B》2021年第2期164-175,共12页中国物理B（英文版）

基　　金：Project supported by the National Natural Science Foundation of China(Grant Nos.11501495,51779215,and 11672259);the Postdoctoral Science Foundation of China(Grant Nos.2015M581869 and 2015T80589);the Jiangsu Government Scholarship for Overseas Studies,China(Grant No.JS-2017-227)。

摘　　要：A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

关 键 词：Caputo fractional derivative nonlinear Schrodinger/Gross-Pitaevskii equation corrected smoothed particle hydrodynamics irregularly domain 

分 类 号：O241.82[理学—计算数学]