Numerical study of nonlinear Schr&#246;dinger equation with high-order split-step corrected smoothed particle hydrodynamics method

Tao Jiang; Jin-Jing Huang; Lin-Guang Lu; Jin-Lian Ren

doi:10.7498/aps.68.20190169

Journals >Acta Physica Sinica >Volume 68 >Issue 9 >Page 090203-1 > Article

Acta Physica Sinica
Vol. 68, Issue 9, 090203-1 (2019)

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

Tao Jiang, Jin-Jing Huang, Lin-Guang Lu, and Jin-Lian Ren^*

DOI: 10.7498/aps.68.20190169 Cite this Article

Tao Jiang, Jin-Jing Huang, Lin-Guang Lu, Jin-Lian Ren. Numerical study of nonlinear Schrödinger equation with high-order split-step corrected smoothed particle hydrodynamics method[J]. Acta Physica Sinica, 2019, 68(9): 090203-1 Copy Citation Text

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Fig. 1. Curve of the along x-axis at different time with : (a) ; (b) . 时不同时刻的实部沿x轴的变化　(a) ; (b)

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Fig. 2. Curve of the along x-axis at different time with : (a) t = 0.1; (b) t = 1. 时不同时刻的实部沿x轴的变化　(a) t = 0.1; (b) t = 1

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Fig. 3. Solitary wave propagation process of at different time with the initial condition 1: (a) ; (b) ; (c) ; (d) . 初始条件1下, 在4个不同时刻孤立波函数的传播过程　(a) ; (b) ; (c) ; (d)

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Fig. 4. Solitary wave propagation process of at different time with initial condition 2: (a) ; (b) ; (c) ; (d) . 初始条件2下, 在4个不同时刻三孤立子波函数的传播过程　(a) ; (b) ; (c) ; (d)

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Fig. 5. Contours of obtained using different methods at two different times: (a) ; (b) . 在两个不同时刻不同数值方法得到的等值线图 (a) ; (b)

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Fig. 6. Curve of along y-axis at different time. 不同时刻沿y轴变化曲线

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Fig. 7. Contour of along different profile at different time: (a) ; (b) . 在3个不同时刻在不同截面上的等值线　(a) 截面; (b) 截面

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Fig. 8. Curve of along x-axis ( = 0.5) at two different time: (a) = 0.05; (b) = 0.25. 两个不同时刻下沿轴( = 0.5)的变化　(a) = 0.05; (b) = 0.25

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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) ; (b1), (b2) ; (c1), (c2) . 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) ; (b1), (b2) ; (c1), (c2)

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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

Table 1. Error obtained using two different methods at different time ( ). 时几个不同时刻里两种方法的误差

	$h = {\text{π}}/32$		$h = {\text{π}}/64$		$h = {\text{π}}/128$
	${e_{\rm{m}}}$	$o{r_{\rm{\alpha }}}$	${e_{\rm{m}}}$	$o{r_{\rm{\alpha }}}$	${e_{\rm{m}}}$	$o{r_{\rm{\alpha }}}$
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

Table 2. Error and convergent order obtained using two different methods at and different particle distance ( ). , 时间t = 1时, 两种方法在不同粒子间距下的误差和收敛阶

	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

Table 3. Error obtained using different methods at different distribution ( , ). 时, 粒子分布均匀或不均匀方式下, 两种方法的误差

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

Table 4. Error obtained using two different methods at three times ( ). 时, 三个不同时刻两种方法的最大误差

	$h = {\text{π}}/32$		$h = {\text{π}}/64$		$h = {\text{π}}/128$
	${e_{\rm{m}}}$	$o{r_{\rm{\alpha }}}$	${e_{\rm{m}}}$	$o{r_{\rm{\alpha }}}$	${e_{\rm{m}}}$	$o{r_{\rm{\alpha }}}$
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

Table 5.

Error and order of convergence by different methods at t = 1 and different h.

t = 1时不同空间步长情况下两种粒子方法的误差和收敛阶

CPU数量	步数			相对加速比S
CPU数量	num = 1	num = 10	num = 1000	相对加速比S
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

Table 6. Consumed CPU time (unit: s) of different calculated time step with particle number at different CPUs. 粒子数为时, 不同CPU个数下运行到不同步数所需时间(单位: s)

粒子数	CPU数量
粒子数	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

Table 7.

The average consumed CPU time (unit: s) of calculated time step 1000 with different particle number and different CPUs.

在不同粒子数下不同CPU个数下, 运行到1000步时平均每步所消耗时间(单位: s)

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