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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: jtrjl_2007@126.com   

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

1 Lavoie J L, Osler T J and Tremblay R 1976 SIAM Rev. 18 240
2 Podlubny I Fractional Differential Equations (New York: Academic Press)
3 Mao Z P and Karniadakis G E 2018 SIAM J. Numer. Anal. 56 24
4 Hu J H, Wang J G and Nie Y F 2019 Chin. Phys. B 28 100201
5 Narahari Achar B N, Yale Bradley T and Hanneken John W 2013 Adv. Math. Phys. 2013 290216
6 Ray S S 2016 Chin. Phys. B 25 040204
7 Alzaidy J F 2013 Amer. J. Math. Anal. 1 14
8 Zhang J, Zhang X.D and Yang B H 2018 Appl. Math. Comput. 335 305
9 Deng W H 2009 SIAM J. Numer. Anal. 47 204
10 Wang H Q 2005 Appl. Math. Comput. 170 17
11 Bao W.Z, Jaksch D and Markowich P A 2003 J. Comput. Phys. 187 318
12 Bao W Z and Wang H Q 2005 Commun. Math. Sci. 3 57
13 Sulem C and Sulem P L 1990 Appl. Math. Sci.(New York: Spinger)
14 Bao W Z and Wang H.Q 2006 J. Comput. Phys. 217 612
15 EI-Danaf T S, Ramadan M A and Abd Alaal F E I 2012 Nonlinear Dyn. 67 619
16 Wilson J P 2019 Comput. Phys. Commun. 235 279
17 Laskin N 2000 Phys. Lett. A 268 298
18 Laskin N 2002 Phys. Rev. E 66 056108
19 Naber M 2004 J. Math. Phys. 45 3339
20 Iomin A 2011 Chaos, Solitons & Fractals 44 348
21 Chen M, Guo Q, Lu D Q and Hu W 2019 Commun. Nonlinear Sci. Numer. Simulat. 71 73
22 Dong J P and Xu M Y 2008 J. Math. Anal. Appl. 344 1005
23 Li M Z, Ding X H and Xu Q 2018 Adv. Differ. Equ. 318 1687
24 Hicdurmaz B and Ashyralyey A 2016 Comput. Math. Appl. 72 1703
25 Chen X L, Di Y N, Duan J Q and Li D F 2018 Appl. Math. Lett. 84 160
26 Ozkan G 2015 Chin. Phys. B 24 100201
27 Edeki S O, Akinilabi G O and Adeosun S A Mohebbi A and Dehghan M 2009 J. Comput. Appl. Math. 225 124
29 Abdel-Salam E A B, Yousif E A and EI-Aasser M A 2016 Rep. Math. Phys. 77 19
30 Xu Y and Shu C W 2005 J. Comput. Phys. 205 72
31 Aboelenen T 2018 Commun. Nonlinear Sci. Numer. Simulat. 54 428
32 Shivanian E and Jafarabadi A 2017 Numer. Methods Partial Differ. Equ. 33 1043
33 Chen R Y, Nie L R and Chen C Y 2018 Chaos 28 053115
34 Azzouzi F, Triki H and Grelu P H 2015 Appl. Math. Model. 39 1300
35 Herzallah M A E and Gepreel K A 2012 Appl. Math. Model. 36 5678
36 Bhrawy A H and Abdelkawy M A 2015 J. Comput. Phys. 294 462
37 Zhang J W, Li D F and Antoine X 2019 Commun. Comput. Phys. 25 218
38 Khan N Alam, Jamil M and Ara Asmat 2012 ISRN Math. Phys. 2012 197068
39 Chen R Y, Tong L M and Nie L R 2017 Physica A 468 532
40 Lin Y M and Xu C J 2007 J. Comput. Phys. 225 1533
41 Gong Y Z, Wang Q, Wang Y S and Cai J X 2017 J. Comput. Phys. 328 354
42 Garrappa R, Moret I and Popolizio M 2015 J. Comput. Phys. 293 115
43 Zhuang P, Gu Y T, Liu F, Turner I and Yarlagadda P K D V 2011 Int. J. Numer. Methods Eng. 88 1346
44 Shivanian E 2015 Int. J. Numer. Methods Eng. 105 83
45 Basic J, Degiuli N and Ban D 2018 J. Comput. Phys. 354 269
46 Mohebbi A, Abbaszadeh M and Dehghan M 2013 Eng. Anal. Bound. Elem. 37 475
47 Liu G R and Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific)
48 Chen J K and Beraun J E 2000 Comput. Methods Appl. Mech. Eng. 190 225
49 Quinlan N J, Basa M and Lastiwka M 2016 Int. J. Numer. Methods Eng. 66 2064
50 Jiang T, Chen Z C, Lu W G, Yuan J Y and Wang D S 2018 Comput. Phys. Commun. 231 19
51 Jiang T, Huang J J, Lu L G and Ren J L 2019 Acta Phys. Sin. 68 090203 (in Chinese)
52 Li S F and Liu W K 2002 Appl. Mech. Rev. 55 1
53 Dehghan M, Abbaszadeh M and Mohebbi A 2014 Cmes-Comp. Model. Eng. 100 399
54 Tayebi A, Shekari Y and Heydari M H 2017 J. Comput. Phys. 340 655
55 Liu J M, Li X K and Hu X L 2019 J. Comput. Phys. 384 222
56 Liu M B and Liu G R 2010 Arch. Comput. Methods Eng. 17 25
57 Crespo A J C, Dominguez J M, Rogers B D and Gomez-Gesteira M 2015 Comput. Phys. Commun. 187 204
58 Sun P N, Colagrossi A, Marrone S and Zhang A M 2017 Comput. Meth. Appl. Mech. Eng. 315 25
59 Ren J L, Jiang T, Lu W G and Li G 2016 Comput. Phys. Commun. 205 87
60 Monaghan J J and Kocharyan A 1995 Comput. Phys. Commun. 87 225
61 Morris J P, Fox P J and Zhu Y 1997 J. Comput. Phys. 136 214
62 Jiang T, Lu L G and Lu W G 2014 Comput. Mech. 53 977
63 Yang X F, Peng S L and Liu M B 2014 Appl. Math. Model. 38 3822
64 Liu M B, Xie W P and Liu G R 2005 Appl. Math. Model. 29 1252
65 Zhang Z L and Liu M B 2018 Appl. Math. Model. 60 606
66 Gao G H, Sun Z Z and Zhang H W 2014 J. Comput. Phys. 259 33
67 Zhou X F, Wu C J and Guo G C 2018 Phys. Rev. Lett. 120 130402
68 Wang T C, Guo B L and Xu Q B 2013 J. Comput. Phys. 243 382
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