Approximate solutions to the Allen–Cahn equation using rational radial basis functions method

Iranian Journal of Numerical Analysis and Optimization

مقاله 23، دوره 13، شماره 2 - شماره پیاپی 25، شهریور 2023، صفحه 187-204   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.76010.1123

نویسندگان

M. Shiralizadeh^{* 1}؛ A. Alipanah¹؛ M. Mohammadi²

¹Department of Applied Mathematics, University of Kurdistan, Sanandaj, Iran.

²Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.

چکیده

We apply the rational radial basis functions (RRBFs) method to solve the Allen–Cahn (A.C) equation, particularly when the equation has a so-lution with steep front or sharp gradients. We approximate the spatial derivatives by the RRBFs method. Then we apply an explicit, fourth-order Runge–Kutta method to advance the resulting semi-discrete system in time. It is well known that the A.C equation has a nonlinear stability feature, meaning that the free-energy functional is reduced by time. The presented method maintains the total energy reduction property of the A.C equation. In the end, five examples to confirm the efficiency and accuracy
of the proposed method are provided.

کلیدواژه‌ها

Allen–Cahn equation؛ RBFs؛ Rational RBFs method؛ Runge–Kutta method

مراجع

[1] Allen, S.M. and Cahn, J.W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall 27 (1979) 1085–1095.
[2] Benes, M., Chalupecky, V. and Mikula, K. Geometrical image segmenta-tion by the Allen–Cahn equation, Appl. Numer. Math. 51 (2004) 187–205.
[3] Boettinger, W.J., Warren, J.A., Beckermann, C. and Karma, A. Phase-field simulation of solidification, Annu. Rev. Mater. Sci. 32 (2002) 163–194.
[4] De Marchi, S., Martinez, A. and Perracchione, E. Fast and stable rational RBF-based partition of unity interpolation, J. Comput. Appl. Math. 349 (2019) 331–343.
[5] Driscoll, A. and Heryudono, R.H. Adaptive residual subsampling methods for radial basis function interpolation and collocation problems, Comput. Math. Appl. 53 (2007) 927–939.
[6] Elliott, C.M. and Stinner, B. Computation of two-phase biomembranes with phase dependent material parameters using surface finite element, Commun. Comput. Phys. 13 (2013) 325–360.
[7] Elliott, C.M. and Stuart, A.M. The global dynamics of discrete semilin-ear parabolic equations, SIAM J. Numer. Anal. 30 (1993) 1622–1663.
[8] Esedoglu, S. and Tsai, Y.H.R. Threshold dynamics for the piecewise constant Mumford-Shan functional, J. Comput. Phys. 211 (2006) 367–384.
[9] Fan, D. and Chen, L.Q. Computer simulation of grain growth using a continuum field model, Acta Mater. 45 (1997) 611–622.
[10] Fasshauer, G.E. Meshfree Approximation Methods with Matlab, World Scientific, 2007.
[11] Feng, X., Li, Y. and Zhang, Y. Finite element methods for the stochastics Allen–Cahn equation with gradient-type multiplicative noise, SIAM J. Numer. Anal. 55(1) (2017) 194–216.
[12] Feng, X., Song, H., Tang, T. and Yang, J. Nonlinear stability of the implicit-explicit methods for the Allen–Cahn equation, Inverse Probl. Imaging 7 (2013) 679–695.
[13] Golubovic, L., Levandovsky, A. and Moldovan, D. Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: Continuum theory insights, East Asian J. Appl. Math. 1 (2011) 297–371.

[14] Heidari, M., Mohammadi, M. and De Marchi, S. A shape preserv-ing quasi-interpolation operator based on a new transcendental RBF, Dolomites Research Notes on Approximation 14 (1) (2021) 56–73.
[15] Heydari, M.H. and Hosseininia, M. A new variable-order fractional derivative with non-singular Mittag-Leffler kernel: application to variable-order fractional version of the 2D Richard equation, Engineering with Computers 38 (2) (2022) 1759–1770.
[16] Hosseininia, M., Heydari, M.H., Avazzadeh, Z. and Maalek Ghaini, F.M. A hybrid method based on the orthogonal Bernoulli polynomials and radial basis functions for variable order fractional reaction-advection-diffusion equation, Engineering Analysis with Boundary Elements 127(2021) 18–28.
[17] Jakobsson, S., Andersson, B. and Edelvik, F. Rational radial basis func-tion interpolation with applications to antenna design, J. Comput. Appl. Math. 233 (2009) 889–904.
[18] Jafari-Varzaneh, H.A. and Hosseini, S.M. A new map for the Chebyshev pseudospectral solution of differential equations with large gradients, Nu-mer. Algorithms 69 (2015) 95–108.
[19] Jeong, D. and Kim, J. An explicit hybrid finite difference scheme for the Allen–Cahn equation, J. Comput. Appl. Math. 340 (2018) 247–255.
[20] Kay, D.A. and Tomasi, A. Color image segmentation by the vector valued Allen–Cahn phase-field model: A multigrid solution, IEEE Trans. Image Process. 18 (2009) 2330–2339.
[21] Kim, J. Phase-field models for multi-component fluid flows, Commun. Comput. Phys. 12 (2012) 613–661.
[22] Kim, J., Jeong, D., Yang, S. and Choi, Y. A finite difference method for a conservative Allen–Cahn equation on non-flat surfaces, J. Comput. Phys. 334 (2017) 170–181.
[23] Kobayashi, R., Warren, J.A. and Carter, W.C. A continuum model of grain boundaries, Phys. D 140 (2000) 141-150.
[24] Krill, C.E. and Chen, L.Q. Computer simulation of 3-D grain growth using a phase-field model, Acta Mater. 50 (2002) 3057–3073.
[25] Lee, H. and Lee, J. A semi-analytical Fourier spectral method for the Allen–Cahn equation, Comput. Math. Appl. 68 (3) (2014) 174–184.
[26] Lee, H.G., Shin, J. and Lee, J.Y. First and second order operator splitting method for phase-field crystal equation, J. Comput. Phys. 299 (2015) 82–91.

[27] Li, Y. and Kim, J. Multiphase image segmentation using a phase-field model, Comput. Math. Appl. 62 (2011) 737–745.
[28] Li, Y., Lee, H.G. and Kim, J. A fast, robust and accurate operator splitting method for Phase-field simulation of crystal growth, J. Cryst. Growth 321 (2011) 176–182.
[29] Mohammadi, V., Mirzaei, D. and Dehghan, M. Numerical simulation and error estimation of the time-dependent Allen–Cahn equation on sur-faces with radial basis functions, J. Sci. Comput. 79 (2019) 493–516.
[30] Mohammadi, M., Mokhtari, R. and Schaback, R. A meshless method for solving the 2d brusselator reaction-diffusion system, Comput. Model. Eng. Sci. 101 (2014) 113–138.
[31] Naqvi, S.L., Levesley, J. and Ali, S. Adaptive radial basis function for time dependent partial differential equations, J. Prime Res. Math. 13(2017) 90–106.
[32] Niu, J., Xu, M. and Yao, G. An efficient reproducing kernel method for solving the Allen–Cahn equation, Appl. Math. Lett. 89 (2019) 78–84.
[33] Perracchione, E. Rational RBF-based partition of unity method for effi-ciently and accurately approximating 3D objects, Comp. Appl. Math. 37(2018) 4633-4648.
[34] Saberi Zafarghandi, F. and Mohammadi, M. Numerical approximations for the Riesz space fractional advection-dispersion equations via radial basis functions, Appl. Numer. Math. 144 (2019) 59–82.
[35] Sarra, S.A. and Bai, Y. A rational radial basis function method for accu-rately resolving discontinuities and steep gradients, Appl. Numer. Math. 130 (2018) 131–142.
[36] Schaback, R. Kernel-Based Meshless Methods, Gottingen, 2011.
[37] Shiralizadeh, M., Alipanah, A. and Mohammadi, M. Numerical solu-tion of one-dimensional Sine-Gordon equation using rational radial basis functions, J. Math. Model. 10 (3)(2022) 387–405.
[38] Wendland, H. Scattered data approximation, Cambridge University Press, 2004.