August  2013, 7(3): 679-695. doi: 10.3934/ipi.2013.7.679

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

1. 

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

2. 

Institute of Theoretical and Computational Studies & Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China, China, China

Received  November 2012 Revised  June 2013 Published  September 2013

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.
Citation: Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
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show all references

References:
[1]

Acta Metall, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

Special issue on time integration (Amsterdam, 1996), Appl. Numer. Math., 25 (1997), 151-167. doi: 10.1016/S0168-9274(97)00056-1.  Google Scholar

[3]

SIAM J. Numer. Anal., 32 (1995), 797-823. doi: 10.1137/0732037.  Google Scholar

[4]

Discrete Contin. Dyn. Syst. A, 29 (2011), 1367-1391.  Google Scholar

[5]

Multiscale Model. Simul., 6 (2007), 913-936. doi: 10.1137/060660631.  Google Scholar

[6]

IEEE Trans. Image Proc., 16 (2007), 285-291. doi: 10.1109/TIP.2006.887728.  Google Scholar

[7]

SIAM J. Sci. Comput., 35 (2013), A22-A51. doi: 10.1137/110842855.  Google Scholar

[8]

Commun. Comput. Phys., 13 (2013), 1189-1208.  Google Scholar

[9]

Commun. Comput. Phys., 13 (2013), 929-957.  Google Scholar

[10]

Commun. Comput. Phys., 13 (2013), 325-360.  Google Scholar

[11]

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, 1998., , ().   Google Scholar

[12]

East Asian J. Appl. Math., 3 (2013), 59-80. doi: 10.4208/eajam.200113.220213a.  Google Scholar

[13]

East Asian J. Appl. Math., 1 (2011), 297-371. doi: 10.4208/eajam.040411.030611a.  Google Scholar

[14]

Commun. Comput. Phys., 14 (2013), 1001-1026. Google Scholar

[15]

Appl. Numer. Math., 57 (2007), 616-628. doi: 10.1016/j.apnum.2006.07.026.  Google Scholar

[16]

Commun. Comput. Phys., 13 (2013), 1408-1431.  Google Scholar

[17]

SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[18]

Commun. Comput. Phys., 12 (2012), 613-661. doi: 10.4208/cicp.301110.040811a.  Google Scholar

[19]

Int. J. Numer. Methods Fluids, 20(1995), 1137-1151. doi: 10.1002/fld.1650201003.  Google Scholar

[20]

J. Sci. Comp., 16(2001), 29-45. doi: 10.1023/A:1011146429794.  Google Scholar

[21]

European J. Appl. Math., 14 (2003), 713-743. doi: 10.1017/S095679250300528X.  Google Scholar

[22]

Numer. Meth. Part Differ. Equ., 28 (2012), 1893-1915. doi: 10.1002/num.20707.  Google Scholar

[23]

SIAM J. Sci. Comput., 33 (2011), 1395-1414. doi: 10.1137/100812781.  Google Scholar

[24]

41 Springer, Heidelberg, 2011, xvi+470 pp. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[25]

Discrete Contin. Dyn. Syst. A, 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[26]

SIAM J. Numer. Anal., 44 (2006), 1759-1779. doi: 10.1137/050628143.  Google Scholar

[27]

Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070. doi: 10.3934/dcdsb.2009.11.1057.  Google Scholar

[28]

SIAM J. Sci. Comput., 31 (2009), 3042-3063. doi: 10.1137/080738398.  Google Scholar

[29]

Commun. Comput. Phys., 11 (2012), 1261-1278. doi: 10.4208/cicp.300810.140411s.  Google Scholar

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