<trans-title xml:lang="en">Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method</trans-title>

Fig. 1. Comparisons between the present numerical results and analytical solutions with different times under the uniform and local refinement particle distributions.

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Fig. 2. The numerical convergence versus time under different particle numbers.

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Fig. 3. Different cases of particle distributions: (a) Uniform case; (b) local refinement case; (c) non-uniform case.

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Fig. 4. The present numerical results under the uniform and local refinement particle distributions.

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Fig. 5. The numerical results obtained using FDM and LR-FPM at different times with .

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Fig. 6. The present numerical results under uniform and local refinement particle distributions at , t = 0.2 s.

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Fig. 7. The FPM result at .

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Fig. 8. The contour results obtained using the present method and the numerical results in ref.[11] at : (a) Numerical results in [11]; (b)−(d) present numerical results

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Fig. 9. The numerical convergence obtained using the present method under different particle distributions at .

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Table 1. The L₂-norm error and convergence rate at .
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Table 1. The L₂-norm error and convergence rate at .
粒子间距误差E₂ 收敛阶
${d_0} = {\text{π}}/16$ 1.9623 × 10^–4 —
${d_0} = {\text{π}}/32$ 4.8081 × 10^–5 2.03
${d_0} = {\text{π}}/64$ 1.0688 × 10^–5 2.16

Table 2. The L₂-norm error at different times under the uniform and local refinement particle distributions.
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Table 2. The L₂-norm error at different times under the uniform and local refinement particle distributions.
$t$ 均匀分布局部加密
0.1 2.2976 × 10^–5 9.7058 × 10^–6
0.3 3.4419 × 10^–5 2.5119 × 10^–5
0.5 4.8081 × 10^–5 4.3028 × 10^–5

Table 3. The L₂-norm error with quintic spline kernel and gaussian kernel functions at .
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Table 3. The L₂-norm error with quintic spline kernel and gaussian kernel functions at .
$t$ 五次样条核函数高斯核函数
0.001 0.0082 0.0107
0.005 0.0186 0.0243
0.010 0.0207 0.0272

Table 4. The L₂-norm error and convergence rate at .
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Table 4. The L₂-norm error and convergence rate at .
粒子间距 ${E_2}$ 收敛阶
${d_0} = 1/20$ 0.0332 —
${d_0} = 1/40$ 0.0078 2.09
${d_0} = 1/{\rm{6}}0$ 0.0032 2.20

Table 5. The L₂-norm error at different times under the uniform, local refinement, and non-uniform particle distributions.
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Table 5. The L₂-norm error at different times under the uniform, local refinement, and non-uniform particle distributions.
$t$ 均匀分布局部加密非均匀分布
0.001 0.0082 0.0049 0.0089
0.005 0.0186 0.0124 0.0150
0.010 0.0207 0.0184 0.0233

Table 6. The L₂-norm error and convergence rate at t = 0.01 s under non-uniform particle distribution.
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Table 6. The L₂-norm error and convergence rate at t = 0.01 s under non-uniform particle distribution.
粒子间距 ${E_2}$ 收敛阶
${d_0} = 1/20$ 0.0251 —
${d_0} = 1/30$ 0.0114 1.95
${d_0} = 1/40$ 0.0063 2.06

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Jin-Lian Ren, Rong-Rong Jiang, Wei-Gang Lu, Tao Jiang. Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method[J]. Acta Physica Sinica, 2020, 69(8): 080202-1

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Paper Information

Category:

Received: Dec. 3, 2019

Accepted: --

Published Online: Nov. 24, 2020

The Author Email:

DOI:10.7498/aps.69.20191829

Topics

Nuclear physics

Particles and fields

Particle and nuclear astrophysics and cosmology

Table 1. The L2-norm error and convergence rate at .

Table 1. The L2-norm error and convergence rate at .

Table 2. The L2-norm error at different times under the uniform and local refinement particle distributions.

Table 2. The L2-norm error at different times under the uniform and local refinement particle distributions.

Table 3. The L2-norm error with quintic spline kernel and gaussian kernel functions at .

Table 3. The L2-norm error with quintic spline kernel and gaussian kernel functions at .

Table 4. The L2-norm error and convergence rate at .

Table 4. The L2-norm error and convergence rate at .

Table 5. The L2-norm error at different times under the uniform, local refinement, and non-uniform particle distributions.

Table 5. The L2-norm error at different times under the uniform, local refinement, and non-uniform particle distributions.

Table 6. The L2-norm error and convergence rate at t = 0.01 s under non-uniform particle distribution.

Table 6. The L2-norm error and convergence rate at t = 0.01 s under non-uniform particle distribution.

Table 1. The L₂-norm error and convergence rate at .

Table 1. The L₂-norm error and convergence rate at .

Table 2. The L₂-norm error at different times under the uniform and local refinement particle distributions.

Table 2. The L₂-norm error at different times under the uniform and local refinement particle distributions.

Table 3. The L₂-norm error with quintic spline kernel and gaussian kernel functions at .

Table 3. The L₂-norm error with quintic spline kernel and gaussian kernel functions at .

Table 4. The L₂-norm error and convergence rate at .

Table 4. The L₂-norm error and convergence rate at .

Table 5. The L₂-norm error at different times under the uniform, local refinement, and non-uniform particle distributions.

Table 5. The L₂-norm error at different times under the uniform, local refinement, and non-uniform particle distributions.

Table 6. The L₂-norm error and convergence rate at t = 0.01 s under non-uniform particle distribution.

Table 6. The L₂-norm error and convergence rate at t = 0.01 s under non-uniform particle distribution.