NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

Qianqian Chu1,2, Guanghui Jin1, Jihong Shen2, Yuanfeng Jin1   

  1. 1. Department of Mathematics, Yanbian University, Yanji 133002, China;
    2. Department of Mathematics Science, Harbin Engineering University, Harbin 150001, China
  • Received:2019-09-18 Revised:2020-01-19 Published:2021-10-15
  • Supported by:
    This work was supported by National Natural Science Foundation of China (No. 11761074), the projection of the Department of Science and Technology of Jilin Province for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project.

Qianqian Chu, Guanghui Jin, Jihong Shen, Yuanfeng Jin. NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION[J]. Journal of Computational Mathematics, 2021, 39(5): 655-665.

We consider numerical methods to solve the Allen-Cahn equation using the secondorder Crank-Nicolson scheme in time and the second-order central difference approach in space. The existence of the finite difference solution is proved with the help of Browder fixed point theorem. The difference scheme is showed to be unconditionally convergent in L norm by constructing an auxiliary Lipschitz continuous function. Based on this result, it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size. The numerical experiments also verify the reliability of the method.

CLC Number: 

[1] G.D. Akrivis, Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal., 13(1993), 115-124.
[2] S.M. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall, 27(1979), 1085-1095.
[3] F.E. Browder, Existence and uniqueness theorems for solutions of nonlinear bourdary value problems. Proc. Sympos. Appl. Math., 17(1965), 24-49.
[4] D.S. Cohen, J.D. Murray, A generalized diffusion model for growth and dispersal in a population. J. Math. Biol., 12(1981), 237-249.
[5] L.C. Evans, H.M. Soner, Souganidis, P.E. Phase transitions and generalized motion by mean curvature. Commun. Pure. Appl. Math., 45(1992), 1097-1123.
[6] X. Feng, T. Tang, J. Yang, Stabilized Crank-Nicolson scheme for phase field models. East Asian J. Appl. Math., 3(2013), 59-80.
[7] T.L. Hou, T. Tang, J. Yang, Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations. J. Sci. Comput., 72(2017), 1214-1231.
[8] J. Kim, Phase-field modes for multi-component fluid flows. Commun. Comput. Phys., 12(2012), 613-661.
[9] G.D. Smith, Numerical Solution of Partial Differential Equations:Finite Difference Methods, 3rd ed. Oxford University Press, Oxford, 1996.
[10] Z.Z. Sun, Finite Difference Methods for Nonlinear Evolutionary Differential Equations. Science Press, Beijing, 2018.
[11] Z.Z. Sun, Numerical Solutions of Partial Differential Equations, 2rd ed. Science Press, Beijing, 2012
[12] T. Tang, Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle. J. Comput. Math., 34:5(2016), 451-461.
[13] J.Q. Zhang, T.L. Hou, Discrete maximum principle and energy stability of finite difference methods for one-dimensional Allen-Cahn equations. Journal of Beihua University, 17:2(2016), 159-164.
[14] N. Zheng, S.Y. Zhai, Z.F. Weng, Two efficient numerical schemes for the Allen-Cahn equation. Adv. in Appl. Math., 6:3(2017), 283-295.
[1] Yong Liu, Chi-Wang Shu, Mengping Zhang. SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS [J]. Journal of Computational Mathematics, 2021, 39(4): 518-537.
[2] Xiaobing Feng, Yukun Li, Yi Zhang. STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE [J]. Journal of Computational Mathematics, 2021, 39(4): 574-598.
[3] Bin Huang, Aiguo Xiao, Gengen Zhang. IMPLICIT-EXPLICIT RUNGE-KUTTA-ROSENBROCK METHODS WITH ERROR ANALYSIS FOR NONLINEAR STIFF DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(4): 599-620.
[4] Yaozong Tang, Qingzhi Yang, Gang Luo. CONVERGENCE ANALYSIS ON SS-HOPM FOR BEC-LIKE NONLINEAR EIGENVALUE PROBLEMS [J]. Journal of Computational Mathematics, 2021, 39(4): 621-632.
[5] Yuting Chen, Mingyuan Cao, Yueting Yang, Qingdao Huang. AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS [J]. Journal of Computational Mathematics, 2021, 39(3): 358-374.
[6] Keke Zhang, Hongwei Liu, Zexian Liu. A NEW ADAPTIVE SUBSPACE MINIMIZATION THREE-TERM CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION [J]. Journal of Computational Mathematics, 2021, 39(2): 159-177.
[7] Huaijun Yang, Dongyang Shi, Qian Liu. SUPERCONVERGENCE ANALYSIS OF LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHODS FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 63-80.
[8] Yongtao Zhou, Chengjian Zhang, Huiru Wang. BOUNDARY VALUE METHODS FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 108-129.
[9] Xiaocui Li, Xu You. MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 130-146.
[10] Wenqi Tao, Zuoqiang Shi. CONVERGENCE OF LAPLACIAN SPECTRA FROM RANDOM SAMPLES [J]. Journal of Computational Mathematics, 2020, 38(6): 952-984.
[11] Wenjuan Xue, Weiai Liu. A MULTIDIMENSIONAL FILTER SQP ALGORITHM FOR NONLINEAR PROGRAMMING [J]. Journal of Computational Mathematics, 2020, 38(5): 683-704.
[12] Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK [J]. Journal of Computational Mathematics, 2020, 38(5): 748-767.
[13] Mohammad Tanzil Hasan, Chuanju Xu. HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY [J]. Journal of Computational Mathematics, 2020, 38(4): 580-605.
[14] Liying Zhang, Jing Wang, Weien Zhou, Landong Liu, Li Zhang. CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(3): 487-501.
[15] Yu Du, Haijun Wu, Zhimin Zhang. SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(1): 223-238.
Viewed
Full text


Abstract