Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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HIGHER ORDER OPERATOR SPLITTING FOURIER SPECTRAL METHODS FOR THE ALLEN-CAHN EQUATION

  • SHIN, JAEMIN (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY) ;
  • LEE, HYUN GEUN (DEPARTMENT OF MATHEMATICS, KWANGWOON UNIVERSITY) ;
  • LEE, JUNE-YUB (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
  • Received : 2017.02.10
  • Accepted : 2017.02.23
  • Published : 2017.03.25
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Abstract

The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. The purpose of this paper is to characterize higher order operator splitting schemes and propose several higher order methods. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095. https://doi.org/10.1016/0001-6160(79)90196-2
  2. M. Benes, V. Chalupecky and K. Mikula, Geometrical image segmentation by the Allen-Cahn equation, Applied Numerical Mathematics, 51 (2004), 187-205. https://doi.org/10.1016/j.apnum.2004.05.001
  3. J.A. Dobrosotskaya and A.L. Bertozzi, A wavelet-Laplace variational technique for image deconvolution and inpainting, IEEE Transactions on Image Processing, 17 (2008), 657-663. https://doi.org/10.1109/TIP.2008.919367
  4. L.C. Evans, H.M. Soner and P.E. Souganidis, Phase transitions and generalized motion by mean curvature, Communications on Pure and Applied Mathematics, 45 (1992), 1097-1123. https://doi.org/10.1002/cpa.3160450903
  5. X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numerische Mathematik, 94 (2003), 33-65. https://doi.org/10.1007/s00211-002-0413-1
  6. M. Katsoulakis, G.T. Kossioris and F. Reitich, Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions, Journal of Geometric Analysis, 5 (1995), 255-279. https://doi.org/10.1007/BF02921677
  7. R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Journal of Physics D, 63 (1993), 410-423. https://doi.org/10.1016/0167-2789(93)90120-P
  8. W.J. Boettinger, J.A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194. https://doi.org/10.1146/annurev.matsci.32.101901.155803
  9. A. Karma and W.-J. Rappel, Quantitative phase-field modeling of dendritic growth in two and three dimensions, Physical Review E, 57 (1998), 4323-4349. https://doi.org/10.1103/PhysRevE.57.4323
  10. D.J. Eyre, An unconditionally stable one-step scheme for gradient systems, http://www.math.utah.edu/-eyre/research/methods/stable.ps.
  11. S. Zhai, X. Feng and Y. He, Numerical simulation of the three dimensional Allen-Cahn equation by the highorder compact ADI method, Computer Physics Communications, 185 (2014), 2449-2455. https://doi.org/10.1016/j.cpc.2014.05.017
  12. X. Feng, H. Song, T. Tang and J. Yang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Problems and Imaging, 7 (2013), 679-695. https://doi.org/10.3934/ipi.2013.7.679
  13. A. Christlieb, J. Jones, K. Promislow, B. Wetton and M. Willoughby, High accuracy solutions to energy gradient flows from material science models, Journal of Computational Physics, 257 (2014), 193-215. https://doi.org/10.1016/j.jcp.2013.09.049
  14. A.-K. Kassam and L.N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. https://doi.org/10.1137/S1064827502410633
  15. H.G. Lee and J.-Y. Lee, A semi-analytical Fourier spectral method for the Allen-Cahn equation, Computers & Mathematics with Applications, 68 (2014), 174-184. https://doi.org/10.1016/j.camwa.2014.05.015
  16. Y. Li, H.G. Lee, D. Jeong and J. Kim, An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation, Computers & Mathematics with Applications, 60 (2010), 1591-1606. https://doi.org/10.1016/j.camwa.2010.06.041
  17. S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Applied Numerical Mathematics, 54 (2005), 23-37. https://doi.org/10.1016/j.apnum.2004.10.005
  18. D. Goldman and T.J. Kaper, Nth-order operator splitting schemes and nonreversible systems, SIAM Journal on Numerical Analysis, 33 (1996), 349-367. https://doi.org/10.1137/0733018
  19. H.G. Lee and J. Kim, An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations, Computer Physics Communications, 183 (2012), 2107-2115. https://doi.org/10.1016/j.cpc.2012.05.013
  20. H.G. Lee, J. Shin and J.-Y. Lee, First and second order operator splitting methods for the phase field crystal equation, Journal of Computational Physics, 299 (2015), 82-91. https://doi.org/10.1016/j.jcp.2015.06.038
  21. H.G. Lee and J.-Y. Lee, A second order operator splitting method for Allen-Cahn type equations with nonlinear source terms, Physica A, 432 (2015), 24-34. https://doi.org/10.1016/j.physa.2015.03.012
  22. G.M. Muslu and H.A. Erbay, Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation, Mathematics and Computers in Simulation, 67 (2005), 581-595. https://doi.org/10.1016/j.matcom.2004.08.002
  23. G. Strang, On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis, 5 (1968), 506-517. https://doi.org/10.1137/0705041
  24. R. McLachlan, Symplectic integration of Hamiltonian wave equations, Numerische Mathematik, 66 (1994), 465-492.
  25. N. Ahmed, T. Natarajan and K.R. Rao, Discrete cosine transform, IEEE Transactions on Computers, C-23 (1974), 90-93. https://doi.org/10.1109/T-C.1974.223784
  26. P.C. Fife, Models for phase separation and their mathematics, Electronic Journal of Differential Equations, 2000 (2000), 1-26.