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The phase separation phenomenon between different matters plays an important role in many science fields. And the high order nonlinear Cahn-Hilliard (C-H) equation is often used to describe the phase separation phenomenon between different matters. However, it is difficult to solve the high-order nonlinear C-H equations by the theorical methods and the grid-based methods. Therefore, in this work the meshless methods are addressed, and a local refinement finite pointset method (LR-FPM) is proposed to numerically investigate the high-order nonlinear C-H equations with different boundary conditions. Its constructive process is as follows. 1) The fourth derivative is decomposed into two second derivatives, and then the spatial derivative is discretized by FPM based on the Taylor series expansion and weighted least square method. 2) The local refinement and quintic spline kernel function are employed to improve the numerical accuracy. 3) The Neumann boundary condition with high-order derivatives is accurately imposed when solving the local linear equation sets. The 1D/2D C-H equations with different boundary conditions are first solved to show the ability of the LR-FPM, and the analytical solutions are available for comparison. Meanwhile, we also investigate the numerical error and convergence order of LR-FPM with uniform/non-uniform particle distribution and local refinement. Finally, 1D/2D C-H equation without analytical solution is predicted by using LR-FPM, and compared with the FDM result. The numerical results show that the implement of the boundary value condition is accurate, the LR-FPM indeed has a higher numerical accuracy and convergence order, is more flexible and applicable than the grid-based FDM, and can accurately predict the time evolution of nonlinear diffusive phase separation phenomenon between different materials time.
[1] Wodo O, Ganapathysubramanian B 2011 J. Comput. Phys. 230 6037
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[14] Ye X D, Cheng X L 2005 Appl. Math. Comput. 171 345
[15] De Mello E V L, Filho O T D 2005 Physica A 347 429
Google Scholar
[16] Chen R Y, Pan W L, Zhang J Q, Nie L R 2016 Chaos 26 093113
Google Scholar
[17] Chen R Y, Nie L R, Chen C Y 2018 Chaos 28 053115
Google Scholar
[18] Chen R Y, Nie L R, Chen C Y, Wang C J 2017 J. Stat.Mech: Theory Exp. 2017 013201
Google Scholar
[19] Chen C Y, Chen R Y, Nie L R, Wang C J, Jia Y J 2018 Physica A 491 399
Google Scholar
[20] Abbaszadeh M, Khodadadian A, Parvizi M, Dehghan M, Heitzinger C 2019 Eng. Anal. Boundary Elem. 98 253
Google Scholar
[21] Zhang Z R, Qiao Z H 2012 Commun. Comput. Phys. 11 1261
Google Scholar
[22] Cheng R J, Cheng Y M 2016 Chin. Phys. B 25 020203
Google Scholar
[23] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Mesh-free Particle Method (Singapore: World Scientific) pp35–83
[24] Yang X F, Liu M B 2017 Commun. Comput. Phys. 22 1015
Google Scholar
[25] 杨秀峰, 刘谋斌 2017 物理学报 66 164701
Google Scholar
Yang X F, Liu M B 2017 Acta Phys. Sin. 66 164701
Google Scholar
[26] Sun P N, Colagrossi A, Marrone S, Zhang A M 2017 Comput. Meth. Appl. Mech. Eng. 315 25
Google Scholar
[27] 蒋涛, 黄金晶, 陆林广, 任金莲 2019 物理学报 68 090203
Google Scholar
Jiang T, Huang J J, Lu L G, Ren J L 2019 Acta Phys. Sin. 68 090203
Google Scholar
[28] Suchde P, Kuhnert J, Tiwari S 2018 Comput. Fluids 165 1
Google Scholar
[29] Resédiz-Flores E O, Kuhnert J, Saucedo-Zendejo F R 2018 Eur. J. Appl. Math. 29 450
Google Scholar
[30] Resendiz-Flores E O, Garcia-Calvillo I D 2014 Int. J. Heat Mass Transfer 71 720
Google Scholar
[31] 任金莲, 任恒飞, 陆伟刚, 蒋涛 2019 物理学报 68 140203
Google Scholar
Ren J L, Ren H F, Lu W G, Jiang T 2019 Acta Phys. Sin. 68 140203
Google Scholar
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图 3 不同粒子分布 (a) 粒子均匀分布; (b) 粒子局部加密分布; (c) 粒子非均匀分布
Figure 3. Different cases of particle distributions: (a) Uniform case; (b) local refinement case; (c) non-uniform case.
图 1 几个不同时刻下均匀分布、局部加密情况下的数值解和解析解
Figure 1. Comparisons between the present numerical results and analytical solutions with different times under the uniform and local refinement particle distributions.
图 2 不同粒子数下的收敛速度随时间的变化
Figure 2. The numerical convergence versus time under different particle numbers.
图 4 均匀分布与局部加密情况下的数值结果
Figure 4. The present numerical results under the uniform and local refinement particle distributions.
图 5
${\varepsilon ^{\rm{2}}}{\rm{ = 0}}{\rm{.3}}$ 时不同时刻FDM结果与LR-FPM结果Figure 5. The numerical results obtained using FDM and LR-FPM at different times with
${\varepsilon ^{\rm{2}}}{\rm{ = 0}}{\rm{.3}}$ .
图 6
${\varepsilon ^2} = 0.03,\; {\rm{ }}t = 0.2\;{\rm{ s}}$ 时刻下均匀分布与局部加密情况下数值结果对比Figure 6. The present numerical results under uniform and local refinement particle distributions at
${\varepsilon ^2} \!=\! 0.03$ , t = 0.2 s.
图 7
$t = 0.1\;{\rm{ s}}$ 时刻FPM方法模拟结果Figure 7. The FPM result at
$t = 0.1\;{\rm{ s}}$ .
图 9
$t \!=\! 0.{\rm{08\; s}}$ 时刻粒子局部加密分布情况下的数值收敛Figure 9. The numerical convergence obtained using the present method under different particle distributions at
$t = 0.{\rm{08\; s}}$ .表 1
$t = 0.5\;{\rm{ s}}$ 时不同粒子间距情况下的L2-范数误差${E_2}$ 和收敛阶Table 1. The L2-norm error
${E_2}$ and convergence rate at$t = 0.5\;{\rm{ s}}$ .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 
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表 2 不同时刻下粒子均匀分布与局部加密情况下的L2-范数误差
${E_2}$ 对比Table 2. The L2-norm error
${E_2}$ 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
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表 3 初始间距
${d_0} = 0.04$ 情况下五次样条核函数与高斯核函数的L2-范数误差${E_2}$ 对比Table 3. The L2-norm error with quintic spline kernel and gaussian kernel functions at
${d_0} = 0.04$ .$t$ 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
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表 4
$t = 0.01\;{\rm{ s}}$ 时刻下不同粒子间距的L2-范数误差${E_2}$ 和收敛阶Table 4. The L2-norm error
${E_2}$ and convergence rate at$t = 0.01\;{\rm{ s}}$ .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
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表 5 粒子均匀分布、局部加密分布与非均匀分布情况下的L2-范数误差
${E_2}$ 对比Table 5. The L2-norm error
${E_2}$ 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
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表 6 t = 0.01 s时不同粒子间距非均匀分布情况下的L2-范数误差
${E_2}$ 和收敛阶Table 6. The L2-norm error
${E_2}$ 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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[1] Wodo O, Ganapathysubramanian B 2011 J. Comput. Phys. 230 6037
Google Scholar
[2] Gómez H, Calo V M, Bazilevs Y, Hughes T J R 2008 Comput. Meth. Appl. Mech. Eng. 197 4333
Google Scholar
[3] Kästner M, Metsch P, DeBorst R 2016 J. Comput. Phys. 305 360
Google Scholar
[4] Guo J. Wang C, Wise S M, Yue X Y 2016 Commun. Math. Sci 14 489
Google Scholar
[5] Cahn J W, Hilliard J E 1958 J. Chem. Phys. 28 258
Google Scholar
[6] Wang W S, Chen L, Zhou J 2016 J. Sci. Comput. 67 724
Google Scholar
[7] 鲁百年, 张瑞凤 1997 工程数学学报 14 52
Lu B N, Zhang R F 1997 J. Eng. Math. 14 52
[8] Furihata D 2001 Numer. Math. 87 675
Google Scholar
[9] Zhu J Z, Chen L Q, Shen J, Tikare V 1999 Phys. Rev. E 60 3564
Google Scholar
[10] Choi Y, Jeong D, Kim J 2017 Appl. Math. Comput. 293 320
[11] Dehghan M, Mohammadi V 2015 Eng. Anal. Boundary Elem. 51 74
Google Scholar
[12] He Y N, Liu Y X, Tang T 2007 Appl. Numer. Math. 57 616
Google Scholar
[13] Dehghan M, Abbaszadeh M 2017 Eng. Anal. Boundary Elem. 78 49
Google Scholar
[14] Ye X D, Cheng X L 2005 Appl. Math. Comput. 171 345
[15] De Mello E V L, Filho O T D 2005 Physica A 347 429
Google Scholar
[16] Chen R Y, Pan W L, Zhang J Q, Nie L R 2016 Chaos 26 093113
Google Scholar
[17] Chen R Y, Nie L R, Chen C Y 2018 Chaos 28 053115
Google Scholar
[18] Chen R Y, Nie L R, Chen C Y, Wang C J 2017 J. Stat.Mech: Theory Exp. 2017 013201
Google Scholar
[19] Chen C Y, Chen R Y, Nie L R, Wang C J, Jia Y J 2018 Physica A 491 399
Google Scholar
[20] Abbaszadeh M, Khodadadian A, Parvizi M, Dehghan M, Heitzinger C 2019 Eng. Anal. Boundary Elem. 98 253
Google Scholar
[21] Zhang Z R, Qiao Z H 2012 Commun. Comput. Phys. 11 1261
Google Scholar
[22] Cheng R J, Cheng Y M 2016 Chin. Phys. B 25 020203
Google Scholar
[23] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Mesh-free Particle Method (Singapore: World Scientific) pp35–83
[24] Yang X F, Liu M B 2017 Commun. Comput. Phys. 22 1015
Google Scholar
[25] 杨秀峰, 刘谋斌 2017 物理学报 66 164701
Google Scholar
Yang X F, Liu M B 2017 Acta Phys. Sin. 66 164701
Google Scholar
[26] Sun P N, Colagrossi A, Marrone S, Zhang A M 2017 Comput. Meth. Appl. Mech. Eng. 315 25
Google Scholar
[27] 蒋涛, 黄金晶, 陆林广, 任金莲 2019 物理学报 68 090203
Google Scholar
Jiang T, Huang J J, Lu L G, Ren J L 2019 Acta Phys. Sin. 68 090203
Google Scholar
[28] Suchde P, Kuhnert J, Tiwari S 2018 Comput. Fluids 165 1
Google Scholar
[29] Resédiz-Flores E O, Kuhnert J, Saucedo-Zendejo F R 2018 Eur. J. Appl. Math. 29 450
Google Scholar
[30] Resendiz-Flores E O, Garcia-Calvillo I D 2014 Int. J. Heat Mass Transfer 71 720
Google Scholar
[31] 任金莲, 任恒飞, 陆伟刚, 蒋涛 2019 物理学报 68 140203
Google Scholar
Ren J L, Ren H F, Lu W G, Jiang T 2019 Acta Phys. Sin. 68 140203
Google Scholar
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