求解Allen-Cahn方程的两种高效数值格式

doi:10.12677/AAM.2017.63034

期刊菜单

求解Allen-Cahn方程的两种高效数值格式
Two Efficient Numerical Schemes for the Allen-Cahn Equation

DOI: 10.12677/AAM.2017.63034, PDF, HTML, XML, 被引量下载: 2,599 浏览: 7,645 国家自然科学基金支持
作者: 郑楠, 翟术英, 翁智峰：华侨大学数学科学学院，福建泉州
关键词: 算子分裂算法；热传导方程；帕德逼近；四阶紧致差分格式；能量递减；Splitting Method； Heat Conduction Equation； Padé Approximation； Fourth–Order Compact Scheme； Energy Decline

摘要: 基于算子分裂思想，本文提出了求解Allen-Cahn方程的两种高效算子分裂格式。首先将此方程分裂为线性项与非线性项两个部分：对线性部分，通过二阶中心差分法与四阶紧致差分法分别进行数值计算，并利用傅里叶方法给出了稳定性的分析；对非线性部分进行精确求解。最后，通过数值算例比较两种算子分裂格式的数值解差异以及运行效率，而且验证了两种格式满足能量递减规律。

Abstract: Based on the idea of operator splitting, this paper proposes two efficient operator splitting schemes for the Allen-Cahn equation. The original equation is divided into linear and nonlinear parts. The linear part is approximated numerically by two different schemes: the second-order center difference scheme and the fourth-order compact difference scheme. The stability analysis of both schemes is discussed according to a Fourier stability analysis. The nonlinear part is solved accurately. Numerical comparisons are carried out to verify the accuracy and efficiency of the proposed methods, and to verify the given schemes satisfied the law of decreasing energy.

文章引用：郑楠, 翟术英, 翁智峰. 求解Allen-Cahn方程的两种高效数值格式[J]. 应用数学进展, 2017, 6(3): 283-295. https://doi.org/10.12677/AAM.2017.63034

参考文献

[1]	Allen, S.M. and Cahn, J.W. (1979) A Microscopie Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening. Acta Metallurgica, 27, 1085-1095.
[2]	Cohen, D.S. and Murray, J.D. (1981) A Generalized Diffusion Model for Growth and Dispersal in a Population. Journal of Mathematical Biology, 12, 237-249. https://doi.org/10.1007/BF00276132
[3]	Hazewinkel, M., Kaashoek, J.F. and Leynse, B. (1986) Pattern Formation for a One Dimensional Evolution Equation Based on Thom’s River Basin Model. Springer, Netherlands.
[4]	Benes, M., Chalupecky, V. and Mikula, K. (2004) Geometrical Image Segmentation by the Allen-Cahn Equation. Applied Numerical Mathematics, 51, 187-205.
[5]	Dobrosotskaya, J.A. and Bertozzi, A.L. (2008) A Wavelet-Laplace Variational Technique for Image Deconvolution and Inpainting. EEE Transactions on Image Processing, 17, 657-663. https://doi.org/10.1109/TIP.2008.919367
[6]	Feng, X.B. and Prohl, A. (2003) Numerical Analysis of the Allen-Cahn Equation and Approximation for Mean Curvature Flows. Numerische Mathematik, 94, 33-65. https://doi.org/10.1007/s00211-002-0413-1
[7]	Wheeler, A.A., Boettinger, W.J. and McFadden, G.B. (1992) Phase-Field Model for Isothermal Phase Transitions in Binary Alloys. Physical Review A, 45, 7424-7439. https://doi.org/10.1103/PhysRevA.45.7424
[8]	Zhai, S., Feng, X. and He, Y. (2014) Numerical Simulation of the Three Dimensional Allen-Cahn Equation by the High-Order Compact ADI Method. Computer Physics Communications, 185, 2449-2455.
[9]	Chertock, A., Doering, C.R., Kashdan, E., et al. (2010) A Fast Explicit Operator Splitting Method for Passive Scalar Advection. Journal of Scientific Computing, 45, 200-214. https://doi.org/10.1007/s10915-010-9381-2
[10]	Dimov, I., Farago, I., Havasi, A. and Zlatev, Z. (2001) L-Commutativity of the Operators of Splitting Methods for Air Pollution Models. Annales Univ. Sci. Budapest Eötvös, Sec. Math., 44, 127-148.
[11]	Farago, I. and Havasi, A. (2007) Consistency Analysis of Operator Splitting Methods for C0-Semigroups Expression. Semigroup Forum, 74, 125-139. https://doi.org/10.1007/s00233-006-0640-3
[12]	Strang, G. (1968) On the Construction and Comparison of Different Splitting Schemes. SIAM Journal on Numerical Analysis, 5, 506-517.
[13]	Strang, G. (1963) Accurate Partial Difference Methods I: Linear Cauchy Problems. Archive for Rational Mechanics and Analysis, 12, 392-402. https://doi.org/10.1007/BF00281235
[14]	Csomos, P., Farago I. and Havasi, A. (2005) Weighted Sequential Splittings and Their Analysis. Computers & Mathematics with Applications, 50, 1017-1031.
[15]	Strang, G. (1968) On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5, 506-517. https://doi.org/10.1137/0705041
[16]	Smith, G.D. (1996) Numerical Solution of Partial Differential Equations (Finite Difference Methods). 3rd Edition, Oxford University Press, Oxford.
[17]	Bagrinovskii, K.A. and Godunov, S.K. (1957) Difference Schemes for Multidimensional Problems. Doklady Akademii Nauk SSSR (NS), 115, 431-433.
[18]	Weng, Z. and Tang, L. (2016) Analysis of the Operator Splitting Scheme for the Allen-Cahn Equation. Journal Numerical Heat Transfer, Part B: Fundamentals, 70, 472-483. https://doi.org/10.1080/10407790.2016.1215714
[19]	Feng, X., Tang, T. and Yang, J. (2015) Long Time Numerical Simulations for Phase-Field Problems Using p-Adaptive Spectral Deferred Correction Methods. SIAM Journal on Scientific Computing, 37, A271-A294. https://doi.org/10.1137/130928662
[20]	Zhai, S., Feng, X. and He, Y. (2014) An Unconditionally Stable Compact ADI Method for Three-Dimensional Time- Fractional Convection-Diffusion Equation. Journal of Computational Physics, 269, 138-155.

为你推荐

友情链接