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Numerical study of nonlinear Schrödinger equation with high-order split-step corrected smoothed particle hydrodynamics method

Jiang Tao, Huang Jin-Jing, Lu Lin-Guang, Ren Jin-Lian
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• 摘要

为提高传统光滑粒子动力学(SPH)方法求解高维非线性薛定谔(nonlinear Schrödinger/Gross-Pitaevskii equation, NLS/GP)方程的数值精度和计算效率, 本文首先基于高阶时间分裂思想将非线性薛定谔方程分解成线性导数项和非线性项, 其次拓展一阶对称SPH方法对复数域上线性导数部分进行显式求解, 最后引入MPI并行技术, 结合边界施加虚粒子方法给出一种能够准确、高效地求解高维NLS/GP方程的高阶分裂修正并行SPH方法. 数值模拟中, 首先对带有周期性和Dirichlet边界条件的NLS方程进行求解, 并与解析解做对比, 准确地得到了周期边界下孤立波的奇异性, 且对提出方法的数值精度、收敛速度和计算效率进行了分析; 随后, 运用给出的高阶分裂粒子方法对复杂二维和三维NLS/GP问题进行了数值预测, 并与其他数值结果进行比较, 准确地展现了非线性孤立波传播中的奇异现象和玻色-爱因斯坦凝聚态中带外旋转项的量子涡旋变化过程.

Abstract

To improve the numerical accuracy and computational efficiency of solving high-dimensional nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation by using traditional SPH method, a high-order split-step coupled with a corrected parallel SPH (HSS-CPSPH) method is developed by applying virtual particles to the boundary. The improvements are described as follows. Firstly, the nonlinear Schrödinger equation is divided into linear derivative term and nonlinear term based on the high-order split-step method. Then, the linear derivative term is solved by extending the first-order symmetric SPH method in explicit time integration. Meanwhile, the MPI parallel technique is introduced to enhance the computational efficiency. In this work, the accuracy, convergence and the computational efficiency of the proposed method are tested by solving the NLS equations with the periodic and Dirichlet boundary conditions, and compared with the analytical solutions. Also, the singularity of solitary waves under the periodic boundary condition is accurately obtained using the proposed particle method. Subsequently, the proposed HSS-CPSPH method is used to predict the results of complex two-dimensional and three-dimensioanl GP problems which are compared with other numerical results. The phenomenon of singular sharp angle in the propagation of nonlinear solitary wave and the process of quantum vortex under Bose-Einstein condensates with external rotation are presented accurately.

作者及机构信息

通信作者: 任金莲, rjl20081223@126.com
• 基金项目: 国家自然科学基金(批准号: 11501495, 51779215)、中国博士后科学基金(批准号: 2015M581869, 2015T80589)、江苏省自然科学基金(批准号: BK20150436)、国家科技支撑计划(批准号: 2015BAD24B02-02)、江苏高校品牌专业建设工程(批准号: PPZY2015B109)和江苏省大学生科技创新项目 (批准号: 201611117016Z) 资助的课题.

Authors and contacts

Corresponding author: Ren Jin-Lian, rjl20081223@126.com
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215), the China Postdoctoral Science Foundation of China (Grant Nos. 2015M581869, 2015T80589), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20150436), the National Key Technologies Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2015BAD24B02-02), the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions, China (Grant No. PPZY2015B109), and the Undergraduate Research and Innovation Project of Jiangsu Province, China (Grant No. 201611117016Z).

施引文献

• 图 1  ${k_{\rm{1}}} = {k_{\rm{2}}} = 1,\;h = {\text{π}}/64$时不同时刻$u\left( {x,{\text{π}}} \right)$的实部沿x轴的变化　(a) $t=1$; (b) $t = 3$

Fig. 1.  Curve of the $\operatorname{Re} \left( {u\left( {x,{\text{π}}} \right)} \right)$ along x-axis at different time with ${k_{\rm{1}}} = {k_{\rm{2}}} = 1,\;h = {\text{π}}/64$: (a) $t=1$; (b) $t = 3$.

图 2  ${k_{\rm{1}}} = {k_{\rm{2}}} = 4,\;h = {\text{π}}/128$时不同时刻$u\left( {x,{\text{π}}} \right)$的实部沿x轴的变化　(a) t = 0.1; (b) t = 1

Fig. 2.  Curve of the $\operatorname{Re} \left( {u\left( {x,{\text{π}}} \right)} \right)$ along x-axis at different time with ${k_{\rm{1}}} = {k_{\rm{2}}} = 4,\;h = {\text{π}}/128$: (a) t = 0.1; (b) t = 1.

图 4  初始条件2下, 在4个不同时刻三孤立子波函数$\left| u \right|$的传播过程　(a) $t=0$; (b) $t = 20$; (c) $t = 30$; (d) $t = 50$

Fig. 4.  Solitary wave propagation process of $\left| u \right|$ at different time with initial condition 2: (a) $t=0$; (b) $t = 20$; (c) $t = 30$; (d) $t = 50$.

图 3  初始条件1下, 在4个不同时刻孤立波函数$\left| u \right|$的传播过程　(a) $t = 0$; (b) $t = 20$; (c) $t = 30$; (d) $t = 50$

Fig. 3.  Solitary wave propagation process of $\left| u \right|$ at different time with the initial condition 1: (a) $t = 0$; (b) $t = 20$; (c) $t = 30$; (d) $t = 50$.

图 5  在两个不同时刻不同数值方法得到的$\left| u \right|$等值线图 (a) $t = 0$; (b) $t = 0.108$

Fig. 5.  Contours of $\left| u \right|$ obtained using different methods at two different times: (a) $t = 0$; (b) $t = 0.108$.

图 6  不同时刻${\left| u \right|^2}$沿y$\left( {x = 0,z = 0} \right)$变化曲线

Fig. 6.  Curve of ${\left| u \right|^2}$ along y-axis $\left( {x = 0,z = 0} \right)$ at different time.

图 7  在3个不同时刻${\left| u \right|^2}$在不同截面上的等值线　(a) $\left( {0,y,z} \right)$截面; (b)$\left( {x,y,0} \right)$截面

Fig. 7.  Contour of ${\left| u \right|^2}$ along different profile at different time: (a) $\left( {0,y,z} \right)$; (b) $\left( {x,y,0} \right)$.

图 8  两个不同时刻下$\left| {{u_1}} \right|$沿$x$轴($y$ = 0.5)的变化　(a) $t$ = 0.05; (b) $t$ = 0.25

Fig. 8.  Curve of $\left| {{u_1}} \right|$ along x-axis ($y$ = 0.5) at two different time: (a) $t$ = 0.05; (b) $t$ = 0.25.

图 9  两个不同时刻下t = 0 (第一列)和t = 0.25 (第二列)三个物理量等值线变化　(a1), (a2) ${\rm Re} ({u_1})$; (b1), (b2) ${\rm Im} ({u_1})$; (c1), (c2) $\left| {{u_1}} \right|$

Fig. 9.  Contours of three physical quantities at two different times t = 0 (the first row) and t = 0.25 (the second row): (a1), (a2) $\operatorname{Re} ({u_1})$; (b1), (b2) $\operatorname{Im} ({u_1})$; (c1), (c2) $\left| {{u_1}} \right|$.

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出版历程
• 收稿日期:  2019-01-29
• 修回日期:  2019-02-24
• 上网日期:  2019-05-01
• 刊出日期:  2019-05-05

非线性薛定谔方程的高阶分裂改进光滑粒子动力学算法

• 扬州大学数学科学学院, 水利与能源动力工程学院, 扬州　225002
• 通信作者: 任金莲, rjl20081223@126.com
基金项目: 国家自然科学基金(批准号: 11501495, 51779215)、中国博士后科学基金(批准号: 2015M581869, 2015T80589)、江苏省自然科学基金(批准号: BK20150436)、国家科技支撑计划(批准号: 2015BAD24B02-02)、江苏高校品牌专业建设工程(批准号: PPZY2015B109)和江苏省大学生科技创新项目 (批准号: 201611117016Z) 资助的课题.

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