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Discrete maximum-norm stability of a linearized second-order finite difference scheme for Allen–Cahn equation

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Abstract

In this paper, we use finite difference methods for solving the Allen–Cahn equation that 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 establish discrete boundedness stability in maximum norm: if the initial data are bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. It is shown that the main result can be obtained under certain restrictions on the time step.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beihua University, Jilin, 132013, China

    T. Hou, K. Wang, Y. Xiong, X. Xiao & Sh. Zhang

Authors
  1. T. Hou
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  2. K. Wang
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  3. Y. Xiong
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  4. X. Xiao
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  5. Sh. Zhang
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Correspondence to T. Hou.

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Original Russian Text © T. Hou, K. Wang, Y. Xiong, X. Xiao, Sh. Zhang, 2017, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2017, Vol. 20, No. 2, pp. 215–222.

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Hou, T., Wang, K., Xiong, Y. et al. Discrete maximum-norm stability of a linearized second-order finite difference scheme for Allen–Cahn equation. Numer. Analys. Appl. 10, 177–183 (2017). https://doi.org/10.1134/S1995423917020082

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  • DOI: https://doi.org/10.1134/S1995423917020082

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