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Journal of Scientific Computing

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25-02-2017

Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

Authors: Tianliang Hou, Tao Tang, Jiang Yang

Published in: Journal of Scientific Computing | Issue 3/2017

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Abstract

We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only \(\mathcal {O}(N\log N)\) computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.

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Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)CrossRef

Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 54(4), 937–954 (2014)MathSciNetCrossRefMATH

Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34, A2145–A2172 (2012)MathSciNetCrossRefMATH

Choi, J.W., Lee, H.G., Jeong, D., Kim, J.: An unconditionally gradient stable numerical method for solving the Allen–Cahn equation. Phys. A Stat. Mech. Appl. 388(9), 1791–1803 (2009)MathSciNetCrossRef

Du, Q., Yang, J.: Asymptotic compatible Fourier spectral approximations of nonlocal Allen–Cahn equations. SIAM J. Numer. Anal. 54(3), 1899–1919 (2016)MathSciNetCrossRefMATH

Eyre, D.J.: An unconditionally stable one-step scheme for gradient systems, Unpublished. http://www.math.utah.edu/eyre/research/methods/stable.ps (1998)

Feng, X., Prohl, A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94(1), 33–65 (2003)MathSciNetCrossRefMATH

Feng, X., Song, H., Tang, T., Yang, J.: Nonlinear stability of the implicit–explicit methods for the Allen–Cahn equation. Inverse Probl. Imaging 7, 679–695 (2013)MathSciNetCrossRefMATH

Feng, X., Tang, T., Yang, J.: Stabilized Crank–Nicolson/Adams–Bashforth schemes for phase field models. East Asian J. Appl. Math. 3, 59–80 (2013)MathSciNetCrossRefMATH

10.

Kim, J.: Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12, 613–661 (2012)MathSciNetCrossRef

11.

Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)MathSciNetCrossRefMATH

12.

Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)MathSciNetCrossRefMATH

13.

Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRefMATH

14.

Qiao, Z.H., Zhang, Z.R., Tang, T.: An adaptive time-stepping strategy for the molecular beam epitaxy models. SIAM J. Sci. Comput. 33, 1395–1414 (2011)MathSciNetCrossRefMATH

15.

Shen, J., Tang, T., Yang, J.: On the maximum principle preserving schemes for the generalized Allen–Cahn equation. Commun. Math. Sci. 14(6), 1517–1534 (2016)MathSciNetCrossRefMATH

16.

Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discret. Contin. Dyn. Syst. 28, 1669–1691 (2010)MathSciNetCrossRefMATH

17.

Tang, T., Yang, J.: Implicit–explicit scheme for the Allen–Cahn equation preserves the maximum principle. J. Comput. Math. 34, 471–481 (2016)MathSciNetCrossRefMATH

18.

Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015)MathSciNetCrossRefMATH

19.

Wikipedia of Tartaglia. https://en.wikipedia.org/wiki/Niccolo_Fontana_Tartaglia

20.

Yang, X.: Error analysis of stabilized semi-implicit method of Allen–Cahn equation. Discrete Contin. Dyn. Syst. Ser. B 11(4), 1057–1070 (2009)MathSciNetCrossRefMATH

21.

Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)MathSciNetCrossRefMATH

22.

Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)MathSciNetCrossRefMATH

23.

Zhang, J., Du, Q.: Numerical studies of discrete approximations to the Allen–Cahn equation in the sharp interface limit. SIAM J. Sci. Comput. 31(4), 3042–3063 (2009)MathSciNetCrossRefMATH

Title: Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations
Authors: Tianliang Hou
Tao Tang
Jiang Yang
Publication date: 25-02-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0396-9

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