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Stabilized Exponential-SAV Schemes Preserving Energy Dissipation Law and Maximum Bound Principle for The Allen–Cahn Type Equations

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Abstract

It is well-known that the Allen–Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years, the scalar auxiliary variable (SAV) method and many of its variants have attracted much attention in numerical solutions for gradient flow problems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. However, existing SAV schemes usually fail to preserve the MBP when applied to the Allen–Cahn equation. In this paper, we develop and analyze new first- and second-order stabilized exponential-SAV schemes for a class of Allen–Cahn type equations, which are shown to simultaneously preserve the energy dissipation law and MBP in discrete settings. In addition, optimal error estimates for the numerical solutions are rigorously obtained for both schemes. Extensive numerical tests and comparisons are also conducted to demonstrate the performance of the proposed schemes.

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Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Lili Ju

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Xiao Li

  3. Department of Applied Mathematics & Research Institute for Smart Energy, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Zhonghua Qiao

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  1. Lili Ju
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  2. Xiao Li
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Correspondence to Xiao Li.

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This work is supported by the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics. L. Ju’s work is partially supported by US National Science Foundation grant DMS-2109633 and US Department of Energy grant DE-SC0020270. X. Li’s work is partially supported by the Hong Kong Research Council GRF grant 15300821 and the Hong Kong Polytechnic University internal grants 4-ZZMK and 1-BD8N. Z. Qiao’s work is partially supported by the Hong Kong Research Council RFS grant RFS2021-5S03 and GRF grants 15300417 and 15302919 and the Hong Kong Polytechnic University internal grant 4-ZZKK.

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Ju, L., Li, X. & Qiao, Z. Stabilized Exponential-SAV Schemes Preserving Energy Dissipation Law and Maximum Bound Principle for The Allen–Cahn Type Equations. J Sci Comput 92, 66 (2022). https://doi.org/10.1007/s10915-022-01921-9

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  • DOI: https://doi.org/10.1007/s10915-022-01921-9

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