Abstract:
In this paper, we use finite difference methods for solving the Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three level scheme in time and a second-order finite difference approach in space, and we establish discrete boundedness stability in maximum norm: if the initial data is bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. We will show that the main result can be obtained under certain.
Citation:
T. Hou, K. Wang, Y. Xiong, X. Xiao, Sh. Zhang, “Discrete maximum-norm stability of a linearized second order finite difference scheme for Allen–Cahn equation”, Sib. Zh. Vychisl. Mat., 20:2 (2017), 215–222; Num. Anal. Appl., 10:2 (2017), 177–183
This publication is cited in the following 12 articles:
Min Li, Wenyi Li, Zirui Du, Tianliang Hou, “A maximum bound principle preserving CN/AB finite difference scheme for Riesz space-fractional Allen-Cahn equations with logarithmic free energy”, Adv Cont Discr Mod, 2025:1 (2025)
Saisai Wang, Guang-an Zou, Bo Wang, “An energy-stable least-squares finite element method for solving the Allen–Cahn equation”, International Journal of Computer Mathematics, 2024, 1
Junxiang Yang, Yibao Li, Chaeyoung Lee, Yongho Choi, Junseok Kim, “Fast evolution numerical method for the Allen–Cahn equation”, Journal of King Saud University - Science, 35:1 (2023), 102430
Zhengyuan Song, Dingqi Li, Dongmei Wang, Huanrong Li, “A modified Crank-Nicolson finite difference method preserving maximum-principle for the phase-field model”, Journal of Mathematical Analysis and Applications, 526:2 (2023), 127271
Yingcong Zhou, Tianliang Hou, “Two-grid algorithm of lumped mass finite element approximation for Allen-Cahn equations”, Computers & Mathematics with Applications, 152 (2023), 46
Huanrong Li, Dongmei Wang, “Numerical analysis of energy-stable Crank-Nicolson finite difference schemes for the phase-field equation”, Journal of Mathematical Analysis and Applications, 514:2 (2022), 126169
Dingwen Deng, Zilin Zhao, “Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation”, Computers & Mathematics with Applications, 128 (2022), 249
树华 林, “Discrete Maximum Principle and Energy Stability Analysis of Du Fort-Frankel Scheme for 1D Allen-Cahn Equation”, PM, 12:09 (2022), 1501
M. Olshanskii, X. Xu, V. Yushutin, “A finite element method for Allen-Cahn equation on deforming surface”, Comput. Math. Appl., 90 (2021), 148–158
D. Jeong, Y. Li, Y. Choi, Ch. Lee, J. Yang, J. Kim, “A practical adaptive grid method for the Allen-Cahn equation”, Physica A, 573 (2021), 125975
T. Hou, H. Leng, “Numerical analysis of a stabilized crank-nicolson/adams-bashforth finite difference scheme for allen-cahn equations”, Appl. Math. Lett., 102 (2020), 106150
V. Yushutin, A. Quaini, Sh. Majd, M. Olshanskii, “A computational study of lateral phase separation in biological membranes”, Int. J. Numer. Meth. Biomed., 35:3 (2019), e3181
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