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Chin. Phys. B, 2021, Vol. 30(2): 020202    DOI: 10.1088/1674-1056/abc0e0
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A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region

Tao Jiang(蒋涛)1,†, Rong-Rong Jiang(蒋戎戎)1, Jin-Jing Huang(黄金晶)1, Jiu Ding(丁玖)2, and Jin-Lian Ren(任金莲)1
1 School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China; 2 Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA
Abstract  A local refinement hybrid scheme (LRCSPH-FDM) is proposed to solve the two-dimensional (2D) time fractional nonlinear Schrödinger 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.
Keywords:  Caputo fractional derivative      nonlinear Schrödinger/Gross-Pitaevskii equation      corrected smoothed particle hydrodynamics      irregularly domain  
Published:  29 January 2021
PACS:  02.60.-x (Numerical approximation and analysis)  
  02.70.-c (Computational techniques; simulations)  
  03.65.Ge (Solutions of wave equations: bound states)  
Fund: 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), and the Jiangsu Government Scholarship for Overseas Studies, China (Grant No. JS-2017-227).
Corresponding Authors:  Corresponding author. E-mail:   

Cite this article: 

Tao Jiang(蒋涛), Rong-Rong Jiang(蒋戎戎), Jin-Jing Huang(黄金晶), Jiu Ding(丁玖), and Jin-Lian Ren(任金莲) A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region 2021 Chin. Phys. B 30 020202

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