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In this paper, a split-step finite pointset method (SS-FPM) is proposed and applied to the simulation of the nonlinear Schrödinger/Gross-Pitaevskii equation (NLSE/GPE) with solitary wave solution. The motivation and main idea of SS-FPMisas follows. 1) The nonlinear Schrödinger equation is first divided into the linear derivative term and the nonlinear term based on the time-splitting method. 2) The finite pointset method (FPM) based on Taylor expansion and weighted least square method is adopted, and the linear derivative term is numerically discretized with the help of Wendland weight function. Then the two-dimensional (2D) nonlinear Schrödinger equation with Dirichlet and periodic boundary conditions is simulated, and the numerical solution is compared with the analytical one. The numerical results show that the presented SS-FPM has second-order accuracy even if in the case of non-uniform particle distribution, and is easily implemented compared with the FDM, and its computational error is smaller than those in the existed corrected SPH methods. Finally, the 2D NLS equation with periodic boundary and the two-component GP equation with Dirichlet boundary and outer rotation BEC, neither of which has an analytical solution, are numerically predicted by the proposed SS-FPM. Compared with other numerical results, our numerical results show that the SS-FPM can accurately display the nonlinear solitary wave singularity phenomenon and quantized vortex process.
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Keywords:
- nonlinear solitary wave /
- finite pointset method /
- splitting scheme /
- Gross-Pitaevskii equation
[1] Bao W Z, Chern I L, Lim F Y 2006 J. Comput. Phys. 219 836
Google Scholar
[2] Qu C, Sun K, Zhang C 2015 Phys. Rev. A 91 053630
Google Scholar
[3] Mason P, Aftalion A 2011 Phys. Rev. A 84 033611
Google Scholar
[4] Antoine X, Bao W, Besse C 2013 Comput. Phys. Commun. 184 2621
Google Scholar
[5] Wang D S, Xue Y S, Zhang Z F 2016 Rom. J. Phys. 61 827
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Google Scholar
[7] Wang H 2005 Appl. Math. Comput. 170 17
[8] Gao Y L, Mei L Q 2016 Appl. Num. Math. 109 41
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[9] Blanes S, Casas F, Murua A 2015 J. Comput. Phys. 303 396
Google Scholar
[10] Dehghan M, Taleei A 2010 Comput. Phys. Commun. 181 43
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[11] Wang T C, Guo B L, Xu Q B 2013 J. Comput. Phys. 243 382
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[12] Chen R Y, Pan W L, Zhang J Q, Nie L R 2016 Chaos 26 093113
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[13] Chen R Y, Tong L M, Nie L R, Wang C I, Pan W 2017 Physica A: Statist. Mech. Appl. 468 532
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[14] Chen R Y, Nie L R, Chen C Y 2018 Chaos 28 053115
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[15] Gong Y Z, Wang Q, Wang Y S, Cai J X 2017 J. Comput. Phys. 328 354
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[17] Dehghan M, Mirzaei D 2008 Int. J. Numer. Meth. 76 501
Google Scholar
[18] Abbasbandy S, Roohani Ghehsareh H, Hashim I 2013 Eng. Anal. Bound. Elem. 37 885
Google Scholar
[19] Liu M B, Liu G R 2010 Arch. Comput. Meth. Eng. 17 25
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Google Scholar
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Google Scholar
[21] Huang C, Lei J M, Liu M B, Peng X Y 2015 In. J. Num. Meth. Flu. 78 691
Google Scholar
[22] 蒋涛, 陈振超, 任金莲, 李刚 2017 物理学报 66 130201
Google Scholar
Jiang T, Chen Z C, Ren J L, Li G 2017 Acta Phys. Sin. 66 130201
Google Scholar
[23] Jiang T, Chen Z C, Lu W G, Yuan J Y, Wang D S 2018 Comput. Phys. Commun. 231 19
Google Scholar
[24] Kuhnert J, Tiwari S 2001 Berichte des Fraunhofer ITWMNr.25
[25] Kuhnert J, Tiwari S 2001 Berichte des Fraunhofer ITWMNr.30
[26] Resendiz-Flores E O, Garcia-Calvillo I D 2014 Int. J. Heat Mass Trans. 71 720
Google Scholar
[27] Wendland H 1995 Adv. Comput. Math. 4 389
Google Scholar
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图 1 几个不同时刻处沿
$y = 0.5{\text{π}}$ 的$\left| \psi \right|$ 变化曲线 (a) 粒子均匀分布; (b) 粒子非均匀分布Figure 1. The change curve of
$\left| \psi \right|$ along$y = 0.5{\text{π}}$ at different time: (a) Uniform mode; (b) non-uniform mode.
图 2 两种不同的粒子分布 (a) 均匀粒子分布; (b) 非均匀粒子分布
Figure 2. Two kinds of particle distribution: (a) Uniform mode; (b) non-uniform mode
图 3 两个不同位置不同时刻ψ实部变化曲线图 (a) 沿对角线; (b) 沿
$y = {\text{π}}$ Figure 3. The change curve of real part at two positions with different times: (a) Along the diagonal; (b) along
$y = {\text{π}}$ 
图 4 两个不同时刻波函数
$\left| \psi \right|$ 三维图和等值线图 (a1), (a2) t = 0; (b1), (b2) t = 0.0108Figure 4. The 3D graphs and contour of
$\left| \psi \right|$ at two different times: (a1), (a2) t = 0; (b1), (b2) t = 0.0108.
图 5 两个不同时刻
$\left| \psi \right|$ 沿x轴 (y = 0)变化曲线 (a) t = 0.05; (b) t = 0.25Figure 5. The change curve of
$\left| \psi \right|$ along x-axis (y = 0) at two different times: (a) t = 0.05; (b) t = 0.25.
图 6 两个不同时刻
${\rm{Re}}\left( \psi \right){\rm{,Im}}\left( \psi \right),\left| \psi \right|$ 的三维数值结果 (a1), (a2), (a3) t = 0; (b1), (b2), (b3) t = 0.25Figure 6. Three-dimensional numerical results of
${\rm{Re}}\left( \psi \right){\rm{,Im}}\left( \psi \right),\left| \psi \right|$ at two different times: (a1), (a2), (a3) t = 0; (b1), (b2), (b3) t = 0.25表 1 粒子分布均匀/非均匀两种情况下的最大误差er
Table 1. Maximum error er under uniform/non-uniform particles distribution
t 均匀分布/10–4 非均匀分布/10–4 0.5 2.48 3.22 1.0 4.94 6.12 1.5 7.40 9.26 2.0 9.88 16.50 
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表 2 四种不同方法在t = 2时的数值收敛阶
Table 2. The rate of convergence obtained using four different methods at t = 2
粒子间距 误差 收敛阶 SS-ICPSPH ${\lambda _0} = {\text{π}}/32$ 8.99×10–2 ${\lambda _0} = {\text{π}}/64$ 2.23×10–3 2.007 ${\lambda _0} = {\text{π}}/128$ 5.52×10–4 2.017 SS-FDM ${\lambda _0} = {\text{π}}/16$ 2.016×10–2 ${\lambda _0} = {\text{π}}/32$ 5.045×10–3 1.9986 ${\lambda _0} = {\text{π}}/64$ 1.262×10–3 1.9997 FPM ${\lambda _0} = {\text{π}}/32$ 4.6×10–3 ${\lambda _0} = {\text{π}}/64$ 1.35×10–3 1.768 ${\lambda _0} = {\text{π}}/128$ 3.86×10–4 1.806 SS-FPM ${\lambda _0} = {\text{π}}/32$ 3.95×10–3 ${\lambda _0} = {\text{π}}/64$ 9.88×10–4 2.000 ${\lambda _0} = {\text{π}}/128$ 2.46×10–4 2.006
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表 3 三种不同方法在t = 2时的数值收敛阶
Table 3. The rate of convergence obtained using three different particle methods at t = 2
粒子间距 误差/× 10–4 收敛阶 SS-ICPSPH ${\lambda _0} = {\text{π}}/32$ 75.530 ${\lambda _0} = {\text{π}}/64$ 18.280 2.046 ${\lambda _0} = {\text{π}}/128$ 4.316 2.082 SS-FDM ${\lambda _0} = {\text{π}}/32$ 24.670 ${\lambda _0} = {\text{π}}/64$ 6.634 1.895 ${\lambda _0} = {\text{π}}/128$ 1.725 1.943 SS-FPM ${\lambda _0} = {\text{π}}/32$ 67.840 ${\lambda _0} = {\text{π}}/64$ 16.790 2.015 ${\lambda _0} = {\text{π}}/128$ 4.128 2.024
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[1] Bao W Z, Chern I L, Lim F Y 2006 J. Comput. Phys. 219 836
Google Scholar
[2] Qu C, Sun K, Zhang C 2015 Phys. Rev. A 91 053630
Google Scholar
[3] Mason P, Aftalion A 2011 Phys. Rev. A 84 033611
Google Scholar
[4] Antoine X, Bao W, Besse C 2013 Comput. Phys. Commun. 184 2621
Google Scholar
[5] Wang D S, Xue Y S, Zhang Z F 2016 Rom. J. Phys. 61 827
[6] Wang D S, Shi Y R, Feng W X, Wen L 2017 Physica D 351−352 30
Google Scholar
[7] Wang H 2005 Appl. Math. Comput. 170 17
[8] Gao Y L, Mei L Q 2016 Appl. Num. Math. 109 41
Google Scholar
[9] Blanes S, Casas F, Murua A 2015 J. Comput. Phys. 303 396
Google Scholar
[10] Dehghan M, Taleei A 2010 Comput. Phys. Commun. 181 43
Google Scholar
[11] Wang T C, Guo B L, Xu Q B 2013 J. Comput. Phys. 243 382
Google Scholar
[12] Chen R Y, Pan W L, Zhang J Q, Nie L R 2016 Chaos 26 093113
Google Scholar
[13] Chen R Y, Tong L M, Nie L R, Wang C I, Pan W 2017 Physica A: Statist. Mech. Appl. 468 532
Google Scholar
[14] Chen R Y, Nie L R, Chen C Y 2018 Chaos 28 053115
Google Scholar
[15] Gong Y Z, Wang Q, Wang Y S, Cai J X 2017 J. Comput. Phys. 328 354
Google Scholar
[16] Cheng R J, Cheng Y M 2016 Chin. Phys. B 25 020203
Google Scholar
[17] Dehghan M, Mirzaei D 2008 Int. J. Numer. Meth. 76 501
Google Scholar
[18] Abbasbandy S, Roohani Ghehsareh H, Hashim I 2013 Eng. Anal. Bound. Elem. 37 885
Google Scholar
[19] Liu M B, Liu G R 2010 Arch. Comput. Meth. Eng. 17 25
Google Scholar
[20] 刘谋斌, 常建忠 2010 物理学报 59 3654
Google Scholar
Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654
Google Scholar
[21] Huang C, Lei J M, Liu M B, Peng X Y 2015 In. J. Num. Meth. Flu. 78 691
Google Scholar
[22] 蒋涛, 陈振超, 任金莲, 李刚 2017 物理学报 66 130201
Google Scholar
Jiang T, Chen Z C, Ren J L, Li G 2017 Acta Phys. Sin. 66 130201
Google Scholar
[23] Jiang T, Chen Z C, Lu W G, Yuan J Y, Wang D S 2018 Comput. Phys. Commun. 231 19
Google Scholar
[24] Kuhnert J, Tiwari S 2001 Berichte des Fraunhofer ITWMNr.25
[25] Kuhnert J, Tiwari S 2001 Berichte des Fraunhofer ITWMNr.30
[26] Resendiz-Flores E O, Garcia-Calvillo I D 2014 Int. J. Heat Mass Trans. 71 720
Google Scholar
[27] Wendland H 1995 Adv. Comput. Math. 4 389
Google Scholar
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