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Third-order accurate, large time-stepping and maximum-principle-preserving schemes for the Allen-Cahn equation

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Abstract

We present and evaluate several explicit, large time-stepping algorithms for the Allen-Cahn equation. Our approach incorporates a stabilization technique and uses Taylor series approximations for exponential functions to develop a family of up to third-order parametric Runge–Kutta schemes that maintain fixed-points and maximum principle for any time step \(\tau > 0\). We also introduce a new relaxation technique that eliminates time delay caused by stabilization. To further decrease the stabilization parameter, we utilize an integrating factor with respect to the stiff linear operator and develop a parametric relaxation integrating factor Runge–Kutta (pRIFRK) framework. Compared to existing maximum-principle-preserving (MPP) schemes, the proposed parametric relaxation approaches are free from limiters, cut-off post-processing, exponential decay, or time delay. Linear stability analysis determines that the parametric approaches are A-stable when appropriate parameters are used. In addition, we provide error estimates in the \(l^\infty \)-norm with the help of the MPP property. We demonstrate the high-order temporal accuracy, maximum-principle-preservation, energy stability, and delay-free properties of the proposed schemes through a set of experiments on 1D, 2D, and 3D problems.

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Acknowledgements

The authors would like to thank the editor and the anonymous referees for their constructive comments and suggestions that greatly improved the quality of this research.

Funding

This work was supported by the National Natural Science Foundation of China (12271523, 12071481, 11971481), Defense Science Foundation of China (2021-JCJQ-JJ-0538), National Key R &D Program of China (SQ2020YFA0709803), Science & Technology Innovation Program of Hunan Province (2021RC3082, 2022RC1192), Natural Science Foundation of Hunan (2021JJ20053), and Research fund from College of Science, National University of Defense Technology (2023-lxy-fhjj-002).

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  1. Department of Mathematics, College of Science, National University of Defense Technology, Changsha, Hunan, 410073, China

    Hong Zhang, Xu Qian & Songhe Song

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  1. Hong Zhang
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H. Zhang: conceptualization, formal analysis, writing—review and editing. X. Qian, S. Song: writing—review and editing.

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Appendix

Appendix

1.1 Maximum-principle-preserving ETD1/2 schemes

Let \(L_\kappa = L - \kappa I\), the stabilization ETD1 and ETD2 schemes [15, 18] for (10) have the forms

$$\begin{aligned} \text {ETD1:~} \varvec{u}^{n+1} = \varphi _0(\tau L_\kappa ) \varvec{u}^n + \tau \varphi _1(\tau L_\kappa ) f_\kappa (\varvec{u}^n), \end{aligned}$$
(36)
$$\begin{aligned} \text {ETD2:~} \left\{ \begin{aligned} \varvec{u}_{n, 1}&= \varphi _0(\tau L_\kappa ) \varvec{u}^n + \tau \varphi _1(\tau L_\kappa ) f_\kappa (\varvec{u}^n), \\ \varvec{u}^{n+1}&\!=\! \varphi _0(\tau L_\kappa ) \varvec{u}^n \!+\! \tau [\varphi _1(\tau L_\kappa ) \!-\! \varphi _2(\tau L_\kappa )]f_\kappa (\varvec{u}^n) \!+\! \tau \varphi _2(\tau L_\kappa ) f_\kappa (\varvec{u}_{n, 1}), \end{aligned}\right. \end{aligned}$$
(37)

where the functions \(\varphi _k(z)\) are recurrently defined by

$$\begin{aligned} \varphi _k(z) = \frac{\varphi _{k-1}(z) - \frac{1}{(k-1)!}}{z}, \forall k \ge 1, \quad \text {with~} \varphi _0(z) = \textrm{e}^z. \end{aligned}$$
(38)

The proofs of maximum-principle-preservation of ETD1/2 can be found in [18, 78].

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Zhang, H., Qian, X. & Song, S. Third-order accurate, large time-stepping and maximum-principle-preserving schemes for the Allen-Cahn equation. Numer Algor 95, 1213–1250 (2024). https://doi.org/10.1007/s11075-023-01606-w

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