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Numerical analysis of a stabilized Crank-Nicolson/Adams-Bashforth finite difference scheme for Allen-Cahn equations. (English) Zbl 1524.65348

Summary: In this paper, we consider finite difference method for solving Allen-Cahn equation which contains small perturbation parameters and strong nonlinearity. We use a stabilized second-order Crank-Nicolson/Adams-Bashforth scheme in time and a second-order finite difference approach in space. It is shown that the numerical solutions satisfy discrete maximum principle under reasonable constraints on time step size and coefficient of stability term. Based on the maximum stability, the discrete energy stability is investigated. Two numerical experiments are performed to verify the theoretical results.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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[1] 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)
[2] Choi, J. W.; Lee, H. G.; Jeong, D., An unconditionally gradient stable numerical method for solving the Allen-Cahn equation, Physica A, 388, 9, 1791-1803 (2009)
[3] Feng, X.; Prohl, A., Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94, 1, 33-65 (2003) · Zbl 1029.65093
[4] Feng, X.; Song, H.; Tang, T.; Yang, J., Nonlinearly stable implicit-explicit methods for the Allen-Cahn equation, Inverse Probl. Imaging, 7, 679-695 (2013) · Zbl 1273.65111
[5] Shen, J.; Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst. Ser. A, 28, 1669-1691 (2010) · Zbl 1201.65184
[6] Yang, X., Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11, 4, 1057-1070 (2009) · Zbl 1201.65170
[7] 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) · Zbl 1198.82045
[8] Tang, T.; Yang, J., Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J. Comput. Math., 34, 471-481 (2016)
[9] Shen, J.; Tang, T.; Yang, J., On the maximum principle preserving schemes for the generlized Allen-Cahn equation, Commu. Math. Sci., 14, 6, 1517-1534 (2016) · Zbl 1361.65059
[10] Hou, T.; Tang, T.; Yang, J., Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations, J. Sci. Comput., 72, 3, 1214-1231 (2017) · Zbl 1379.65063
[11] Hou, T.; Wang, K.; Xiong, Y., Discrete maximum-norm stability of a linearized second order finite difference scheme for Allen-Cahn equation, Numer. Anal. Appl., 10, 2, 177-183 (2017) · Zbl 1399.65211
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