A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

Hou, Dianming; Ju, Lili; Qiao, Zhonghua

doi:10.1090/mcom/3843

A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility
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by Dianming Hou, Lili Ju and Zhonghua Qiao

Math. Comp. 92 (2023), 2515-2542

DOI: https://doi.org/10.1090/mcom/3843

Published electronically: May 17, 2023

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Abstract:

In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete $H^{1}$ error estimate and energy stability for the classic constant mobility case and the $L^{\infty }$ error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy.

References

Georgios Akrivis, Buyang Li, and Dongfang Li, Energy-decaying extrapolated RK-SAV methods for the Allen-Cahn and Cahn-Hilliard equations, SIAM J. Sci. Comput. 41 (2019), no. 6, A3703–A3727. MR 4033693, DOI 10.1137/19M1264412
Arvind Baskaran, Zhengzheng Hu, John S. Lowengrub, Cheng Wang, Steven M. Wise, and Peng Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, J. Comput. Phys. 250 (2013), 270–292. MR 3079535, DOI 10.1016/j.jcp.2013.04.024
A. Baskaran, J. S. Lowengrub, C. Wang, and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal. 51 (2013), no. 5, 2851–2873. MR 3118257, DOI 10.1137/120880677
J. Becker, A second order backward difference method with variable steps for a parabolic problem, BIT 38 (1998), no. 4, 644–662. MR 1670200, DOI 10.1007/BF02510406
L. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun. 108 (1998), no. 2–3, 147–158.
Wenbin Chen, Sidafa Conde, Cheng Wang, Xiaoming Wang, and Steven M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput. 52 (2012), no. 3, 546–562. MR 2948706, DOI 10.1007/s10915-011-9559-2
Wenbin Chen, Weijia Li, Zhiwen Luo, Cheng Wang, and Xiaoming Wang, A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection, ESAIM Math. Model. Numer. Anal. 54 (2020), no. 3, 727–750. MR 4080786, DOI 10.1051/m2an/2019054
Wenbin Chen, Cheng Wang, Xiaoming Wang, and Steven M. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, J. Comput. Phys. X 3 (2019), 100031, 29. MR 4116082, DOI 10.1016/j.jcpx.2019.100031
Wenbin Chen, Xiaoming Wang, Yue Yan, and Zhuying Zhang, A second order BDF numerical scheme with variable steps for the Cahn-Hilliard equation, SIAM J. Numer. Anal. 57 (2019), no. 1, 495–525. MR 3916957, DOI 10.1137/18M1206084
Kelong Cheng, Wenqiang Feng, Cheng Wang, and Steven M. Wise, An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation, J. Comput. Appl. Math. 362 (2019), 574–595. MR 3987420, DOI 10.1016/j.cam.2018.05.039
Kelong Cheng, Zhonghua Qiao, and Cheng Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput. 81 (2019), no. 1, 154–185. MR 4002742, DOI 10.1007/s10915-019-01008-y
Kelong Cheng, Cheng Wang, Steven M. Wise, and Yanmei Wu, A third order accurate in time, BDF-type energy stable scheme for the Cahn-Hilliard equation, Numer. Math. Theory Methods Appl. 15 (2022), no. 2, 279–303. MR 4407466, DOI 10.4208/nmtma.oa-2021-0165
Qing Cheng and Jie Shen, A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving, Comput. Methods Appl. Mech. Engrg. 391 (2022), Paper No. 114585, 25. MR 4370503, DOI 10.1016/j.cma.2022.114585
Qing Cheng and Jie Shen, A new Lagrange multiplier approach for constructing structure preserving schemes, II. Bound preserving, SIAM J. Numer. Anal. 60 (2022), no. 3, 970–998. MR 4417002, DOI 10.1137/21M144877X
Jon Matteo Church, Zhenlin Guo, Peter K. Jimack, Anotida Madzvamuse, Keith Promislow, Brian Wetton, Steven M. Wise, and Fengwei Yang, High accuracy benchmark problems for Allen-Cahn and Cahn-Hilliard dynamics, Commun. Comput. Phys. 26 (2019), no. 4, 947–972. MR 3978662, DOI 10.4208/cicp.oa-2019-0006
Qiang Du, Lili Ju, Xiao Li, and Zhonghua Qiao, Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation, SIAM J. Numer. Anal. 57 (2019), no. 2, 875–898. MR 3945242, DOI 10.1137/18M118236X
Qiang Du, Lili Ju, Xiao Li, and Zhonghua Qiao, Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes, SIAM Rev. 63 (2021), no. 2, 317–359. MR 4253790, DOI 10.1137/19M1243750
Zhaohui Fu and Jiang Yang, Energy-decreasing exponential time differencing Runge-Kutta methods for phase-field models, J. Comput. Phys. 454 (2022), Paper No. 110943, 11. MR 4369169, DOI 10.1016/j.jcp.2022.110943
Hector Gomez and Thomas J. R. Hughes, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys. 230 (2011), no. 13, 5310–5327. MR 2799512, DOI 10.1016/j.jcp.2011.03.033
Yonghong Hao, Qiumei Huang, and Cheng Wang, A third order BDF energy stable linear scheme for the no-slope-selection thin film model, Commun. Comput. Phys. 29 (2021), no. 3, 905–929. MR 4203115, DOI 10.4208/cicp.oa-2020-0074
Dianming Hou, Mejdi Azaiez, and Chuanju Xu, A variant of scalar auxiliary variable approaches for gradient flows, J. Comput. Phys. 395 (2019), 307–332. MR 3975040, DOI 10.1016/j.jcp.2019.05.037
D. Hou and Z. Qiao, A linear adaptive BDF2 scheme for phase field crystal equation, arXiv:2206.07625, pages 1–21, 2022.
Dianming Hou and Zhonghua Qiao, An implicit-explicit second-order BDF numerical scheme with variable steps for gradient flows, J. Sci. Comput. 94 (2023), no. 2, Paper No. 39, 22. MR 4530253, DOI 10.1007/s10915-022-02094-1
Dianming Hou and Chuanju Xu, A second order energy dissipative scheme for time fractional L$^2$ gradient flows using SAV approach, J. Sci. Comput. 90 (2022), no. 1, Paper No. 25, 22. MR 4344588, DOI 10.1007/s10915-021-01667-w
Tianliang Hou and Haitao Leng, Numerical analysis of a stabilized Crank-Nicolson/Adams-Bashforth finite difference scheme for Allen-Cahn equations, Appl. Math. Lett. 102 (2020), 106150, 9. MR 4037706, DOI 10.1016/j.aml.2019.106150
Tianliang Hou, Tao Tang, and Jiang Yang, Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations, J. Sci. Comput. 72 (2017), no. 3, 1214–1231. MR 3687899, DOI 10.1007/s10915-017-0396-9
Z. Hu, S. M. Wise, C. Wang, and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys. 228 (2009), no. 15, 5323–5339. MR 2541456, DOI 10.1016/j.jcp.2009.04.020
Kun Jiang, Lili Ju, Jingwei Li, and Xiao Li, Unconditionally stable exponential time differencing schemes for the mass-conserving Allen-Cahn equation with nonlocal and local effects, Numer. Methods Partial Differential Equations 38 (2022), no. 6, 1636–1657. MR 4501939, DOI 10.1002/num.22827
Lili Ju, Xiao Li, and Zhonghua Qiao, Generalized SAV-exponential integrator schemes for Allen-Cahn type gradient flows, SIAM J. Numer. Anal. 60 (2022), no. 4, 1905–1931. MR 4458898, DOI 10.1137/21M1446496
Lili Ju, Xiao Li, and Zhonghua Qiao, Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations, J. Sci. Comput. 92 (2022), no. 2, Paper No. 66, 34. MR 4450101, DOI 10.1007/s10915-022-01921-9
Lili Ju, Xiao Li, Zhonghua Qiao, and Jiang Yang, Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations, J. Comput. Phys. 439 (2021), Paper No. 110405, 18. MR 4258763, DOI 10.1016/j.jcp.2021.110405
Lili Ju, Jian Zhang, Liyong Zhu, and Qiang Du, Fast explicit integration factor methods for semilinear parabolic equations, J. Sci. Comput. 62 (2015), no. 2, 431–455. MR 3299200, DOI 10.1007/s10915-014-9862-9
Buyang Li, Jiang Yang, and Zhi Zhou, Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations, SIAM J. Sci. Comput. 42 (2020), no. 6, A3957–A3978. MR 4186541, DOI 10.1137/20M1333456
Jingwei Li, Lili Ju, Yongyong Cai, and Xinlong Feng, Unconditionally maximum bound principle preserving linear schemes for the conservative Allen-Cahn equation with nonlocal constraint, J. Sci. Comput. 87 (2021), no. 3, Paper No. 98, 32. MR 4257072, DOI 10.1007/s10915-021-01512-0
Jingwei Li, Xiao Li, Lili Ju, and Xinlong Feng, Stabilized integrating factor Runge-Kutta method and unconditional preservation of maximum bound principle, SIAM J. Sci. Comput. 43 (2021), no. 3, A1780–A1802. MR 4259913, DOI 10.1137/20M1340678
Weijia Li, Wenbin Chen, Cheng Wang, Yue Yan, and Ruijian He, A second order energy stable linear scheme for a thin film model without slope selection, J. Sci. Comput. 76 (2018), no. 3, 1905–1937. MR 3833714, DOI 10.1007/s10915-018-0693-y
Xiao Li, Zhonghua Qiao, and Cheng Wang, Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, Math. Comp. 90 (2021), no. 327, 171–188. MR 4166457, DOI 10.1090/mcom/3578
Xiaoli Li, Jie Shen, and Hongxing Rui, Energy stability and convergence of SAV block-centered finite difference method for gradient flows, Math. Comp. 88 (2019), no. 319, 2047–2068. MR 3957886, DOI 10.1090/mcom/3428
Hong-lin Liao, Bingquan Ji, Lin Wang, and Zhimin Zhang, Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model, J. Sci. Comput. 92 (2022), no. 2, Paper No. 52, 26. MR 4448198, DOI 10.1007/s10915-022-01861-4
Hong-lin Liao, William McLean, and Jiwei Zhang, A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal. 57 (2019), no. 1, 218–237. MR 3904430, DOI 10.1137/16M1175742
Hong-lin Liao, Tao Tang, and Tao Zhou, On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen-Cahn equation, SIAM J. Numer. Anal. 58 (2020), no. 4, 2294–2314. MR 4134034, DOI 10.1137/19M1289157
Hong-lin Liao and Zhimin Zhang, Analysis of adaptive BDF2 scheme for diffusion equations, Math. Comp. 90 (2021), no. 329, 1207–1226. MR 4232222, DOI 10.1090/mcom/3585
Chunwan Lv and Chuanju Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput. 38 (2016), no. 5, A2699–A2724. MR 3543161, DOI 10.1137/15M102664X
X. Meng, Z. Qiao, C. Wang, and Z. Zhang, Artificial regularization parameter analysis for the no-slope-selection epitaxial thin film model, CSIAM Trans. Appl. Math. 1 (2020), no. 3, 441–462.
Zhonghua Qiao, Zhengru Zhang, and Tao Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput. 33 (2011), no. 3, 1395–1414. MR 2813245, DOI 10.1137/100812781
Jie Shen, Tao Tang, and Jiang Yang, On the maximum principle preserving schemes for the generalized Allen-Cahn equation, Commun. Math. Sci. 14 (2016), no. 6, 1517–1534. MR 3538360, DOI 10.4310/CMS.2016.v14.n6.a3
Jie Shen, Cheng Wang, Xiaoming Wang, and Steven M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal. 50 (2012), no. 1, 105–125. MR 2888306, DOI 10.1137/110822839
Jie Shen, Jie Xu, and Jiang Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys. 353 (2018), 407–416. MR 3723659, DOI 10.1016/j.jcp.2017.10.021
Jie Shen, Jie Xu, and Jiang Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev. 61 (2019), no. 3, 474–506. MR 3989239, DOI 10.1137/17M1150153
Tao Tang and Jiang Yang, Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J. Comput. Math. 34 (2016), no. 5, 471–481. MR 3549191, DOI 10.4208/jcm.1603-m2014-0017
Alan Weiser and Mary Fanett Wheeler, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal. 25 (1988), no. 2, 351–375. MR 933730, DOI 10.1137/0725025
S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput. 44 (2010), no. 1, 38–68. MR 2647498, DOI 10.1007/s10915-010-9363-4
S. M. Wise, C. Wang, and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2269–2288. MR 2519603, DOI 10.1137/080738143
Yue Yan, Wenbin Chen, Cheng Wang, and Steven M. Wise, A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys. 23 (2018), no. 2, 572–602. MR 3869669, DOI 10.4208/cicp.oa-2016-0197
Jiang Yang, Zhaoming Yuan, and Zhi Zhou, Arbitrarily high-order maximum bound preserving schemes with cut-off postprocessing for Allen-Cahn equations, J. Sci. Comput. 90 (2022), no. 2, Paper No. 76, 36. MR 4362470, DOI 10.1007/s10915-021-01746-y
Hong Zhang, Jingye Yan, Xu Qian, Xianming Gu, and Songhe Song, On the preserving of the maximum principle and energy stability of high-order implicit-explicit Runge-Kutta schemes for the space-fractional Allen-Cahn equation, Numer. Algorithms 88 (2021), no. 3, 1309–1336. MR 4322692, DOI 10.1007/s11075-021-01077-x
Hong Zhang, Jingye Yan, Xu Qian, and Songhe Song, Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation, Appl. Numer. Math. 161 (2021), 372–390. MR 4183772, DOI 10.1016/j.apnum.2020.11.022