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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.
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Keywords:
- nonlinear Schrödinger equation/
- smoothed particle hydrodynamics/
- time split-step/
- Bose-Einstein condensates
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] -
时间t SS-ICPSPH HSS-CPSPH 0.5 1.697 × 10–3 1.696 × 10–3 1 3.616 × 10–3 2.494 × 10–3 2 7.347 × 10–3 4.857 × 10–3 
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$h = {\text{π}}/32$ $h = {\text{π}}/64$ $h = {\text{π}}/128$ ${e_{\rm{m}}}$ $o{r_{\rm{\alpha }}}$ ${e_{\rm{m}}}$ $o{r_{\rm{\alpha }}}$ ${e_{\rm{m}}}$ $o{r_{\rm{\alpha }}}$ SS-ICPSPH 1.381 × 10–2 — 3.616 × 10–3 1.933 9.0412 × 10–4 2.00 HSS-CPSPH 1.381 × 10–2 — 2.494 × 10–3 2.47 4.498 × 10–4 2.47 DownLoad:
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均匀分布粒子 非均匀分布情形1 非均匀分布情形2 $t = 0.1$ $t = 1$ $t = 0.1$ $t = 1$ $t = 0.1$ $t = 1$ SS-ICPSPH 2.776 × 10–4 3.616 × 10–3 2.944 × 10–4 3.818 × 10–3 3.116 × 10–4 4.082 × 10–3 HSS-CPSPH 2.774 × 10–4 2.494 × 10–3 2.886 × 10–4 2.527 × 10–3 2.967 × 10–4 2.578 × 10–3 DownLoad:
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时间t SS-ICPSPH HSS-CPSPH 0.5 9.131 × 10–4 4.512 × 10–4 1 1.828 × 10–3 8.135 × 10–4 2 3.658 × 10–3 1.623 × 10–3 DownLoad:
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$h = {\text{π}}/32$ $h = {\text{π}}/64$ $h = {\text{π}}/128$ ${e_{\rm{m}}}$ $o{r_{\rm{\alpha }}}$ ${e_{\rm{m}}}$ $o{r_{\rm{\alpha }}}$ ${e_{\rm{m}}}$ $o{r_{\rm{\alpha }}}$ SS-ICPSPH 7.553 × 10–3 — 1.828 × 10–3 2.046 4.316 × 10–4 2.082 HSS-CPSPH 4.534 × 10–3 — 8.135 × 10–4 2.476 1.379 × 10–4 2.560 DownLoad:
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CPU数量 步数 相对加速比S num= 1 num= 10 num= 1000 2 97805.9 107508 1174728 — 12 16716.9 18516.7 215526.7 — 24 8388.87 9404.37 120284.37 1.792 36 5603.29 6344.98 87524.98 2.462 72 2948.83 3189.24 48564.28 4.438 DownLoad:
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粒子数 CPU数量 2 12 24 36 72 ${121^3}$ 449.55 82.926 45.962 35.000 19.585 ${161^3}$ 1076.922 198.810 111.90 81.922 47.363 ${181^3}$ 1558.445 292.711 164.838 120.886 65.437 ${201^3}$ 2190.921 425.688 235.775 179.856 96.836 DownLoad:
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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