Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation

Xinlong Feng; Huailing Song; Tao Tang; Jiang Yang

doi:10.3934/ipi.2013.7.679

Article Contents

2013, Volume 7, Issue 3: 679-695. Doi: 10.3934/ipi.2013.7.679

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Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation

1.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046
2.
Institute of Theoretical and Computational Studies & Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Received: November 2012

Revised: June 2013

Published: August 2013

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Abstract

In this paper, we will investigate the first- and second-order implicit-explicit schemes with parameters for solving the Allen-Cahn equation. It is known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. The goal of this paper is to consider implicit-explicit schemes that inherit the nonlinear stability of the continuous model, which will be achieved by properly choosing parameters associated with the implicit-explicit schemes. Theoretical justifications for the nonlinear stability of the schemes will be provided, and the theoretical results will be verified by several numerical examples.

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Mathematics Subject Classification: 65M06, 65M12, 6506, 35k20, 35Q56.

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References

[1]	S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall, 27 (1979), 1085-1095.doi: 10.1016/0001-6160(79)90196-2.
[2]	U. M. Ascher, J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta method for time dependent partial differential equations, Special issue on time integration (Amsterdam, 1996), Appl. Numer. Math., 25 (1997), 151-167.doi: 10.1016/S0168-9274(97)00056-1.
[3]	U. M. Ascher, J. Ruuth and T. R. Wetton, Implicit-explicit method for time dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.doi: 10.1137/0732037.
[4]	A. L. Bertozzi, N. Ju and H.-W. Lu, A biharmonic modified forward time stepping method for fourth order nonlinear diffusion equations, Discrete Contin. Dyn. Syst. A, 29 (2011), 1367-1391.
[5]	A. L. Bertozzi, S. Esedoglu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.doi: 10.1137/060660631.
[6]	A. L. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc., 16 (2007), 285-291.doi: 10.1109/TIP.2006.887728.
[7]	S. Boscarino, L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35 (2013), A22-A51.doi: 10.1137/110842855.
[8]	F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 13 (2013), 1189-1208.
[9]	C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13 (2013), 929-957.
[10]	Charles M. Elliott and Bjorn Stinner, Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements, Commun. Comput. Phys., 13 (2013), 325-360.
[11]	D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, 1998. http://www.math.utah.edu/eyre/research/methods/stable.ps
[12]	X. Feng, T. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80.doi: 10.4208/eajam.200113.220213a.
[13]	L. Golubovic, A. Levandovsky and D. Moldovan, 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.doi: 10.4208/eajam.040411.030611a.
[14]	F. de la Hoz and F. Vadillo, A Sylvester-based IMEX method via differentiation matrices for solving nonlinear parabolic equations, Commun. Comput. Phys., 14 (2013), 1001-1026.
[15]	Y. He, Y. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628.doi: 10.1016/j.apnum.2006.07.026.
[16]	Samet Y. Kadioglu and Dana A. Knoll, A Jacobian-free Newton Krylov implicit-explicit time integration method for incompressible flow problems, Commun. Comput. Phys., 13 (2013), 1408-1431.
[17]	A. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), 1214-1233.doi: 10.1137/S1064827502410633.
[18]	Junseok Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys., 12 (2012), 613-661.doi: 10.4208/cicp.301110.040811a.
[19]	M. Li, T. Tang and B. Fornberg, A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 20(1995), 1137-1151.doi: 10.1002/fld.1650201003.
[20]	M. Li and T. Tang, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, J. Sci. Comp., 16(2001), 29-45.doi: 10.1023/A:1011146429794.
[21]	B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection, European J. Appl. Math., 14 (2003), 713-743.doi: 10.1017/S095679250300528X.
[22]	Z. Qiao, Z. Sun and Z. Zhang, The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model, Numer. Meth. Part Differ. Equ., 28 (2012), 1893-1915.doi: 10.1002/num.20707.
[23]	Z. Qiao, Z. Zhang and T. Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), 1395-1414.doi: 10.1137/100812781.
[24]	J. Shen, T. Tang and L. Wang, "Spectral Methods: Algorithms, Analysis and Applications," 41 Springer, Heidelberg, 2011, xvi+470 pp.doi: 10.1007/978-3-540-71041-7.
[25]	J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst. A, 28 (2010), 1669-1691.doi: 10.3934/dcds.2010.28.1669.
[26]	C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759-1779.doi: 10.1137/050628143.
[27]	X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070.doi: 10.3934/dcdsb.2009.11.1057.
[28]	J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.doi: 10.1137/080738398.
[29]	Z. Zhang and Z. Qiao, An adaptive time-stepping strategy for the Cahn-Hilliard equation, Commun. Comput. Phys., 11 (2012), 1261-1278.doi: 10.4208/cicp.300810.140411s.