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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

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Abstract

We put forward and analyze the high-order (up to fourth) strong stability-preserving implicit-explicit Runge-Kutta schemes for the time integration of the space-fractional Allen-Cahn equation, which inherits the maximum principle preserving and energy stability. The space-fractional Allen-Cahn equation with homogeneous Dirichlet boundary condition is first discretized in the spatial direction by using a second-order fractional centered difference scheme that preserves the semi-discrete maximum principle. It is subsequently integrated in the temporal direction by a class of strong stability-preserving implicit-explicit Runge-Kutta schemes that are specifically designed to preserve the maximum principle to the optimal time step size. The convergence order in the discrete \(L^{\infty }\) norm and energy boundedness are provided by using the established maximum principle. Finally, a series of numerical experiments are carried out to demonstrate the high-order convergence, maximum principle preserving, and energy stability of the proposed schemes.

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Acknowledgements

H. Zhang would like to thank “user1551” in https://math.stackexchange.com/questions/3769352for the proof of Lemma 2.3.

Funding

This work was supported by the National Key R&D Program of China (SQ2020YFA070075), the National Natural Science Foundation of China (No. 11901577, 11971481, 12071481), the Natural Science Foundation of Hunan (No. S2017JJQNJJ0764, 2020JJ5652), the fund from Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD004), the Research Fund of National University of Defense Technology (No. ZK19-37) and the Basic Research Foundation of National Numerical Wind Tunnel Project (No. NNW2018-ZT4A08).

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Authors and Affiliations

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

    Hong Zhang, Jingye Yan, Xu Qian & Songhe Song

  2. School of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, 611130, China

    Xianming Gu

  3. State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan, 410073, China

    Songhe Song

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  1. Hong Zhang
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Appendix:: Coefficients of some IMEX Runge-Kutta methods

Appendix:: Coefficients of some IMEX Runge-Kutta methods

  1. 1.

    Optimized SSP2222 (K = 1)

    $$ \begin{aligned} A & = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right],\\ b & = \left[ \begin{array}{cc} 0.5 & 0.5 \end{array} \right]^{T}, \\ \tilde A & = \left[ \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right] \\ \tilde b &= b. \end{aligned} $$
  2. 2.

    Optimized SSP3333 (K = 1)

    $$ \begin{aligned} A &= \left[ \begin{array}{cccc} 0 & 0 & 0\\ 0.835696619896941& 0 & 0 \\ 0.206998613430894& 0.359713233559212 & 0 \end{array} \right], \\ b &= \left[ \begin{array}{ccc} 0.223244212531201& 0.222329659694216 & 0.554426127774583 \end{array} \right]^{T}, \\ \tilde A &= \left[ \begin{array}{cccc} 0 & 0 & 0\\ 0.626996630922690& 0.208699988974251& 0\\ 0.184038115921842& 0.051324585660262 & 0.331349145408003 \end{array} \right], \\ \tilde b &= b. \end{aligned} $$
  3. 3.

    Optimized SSP4334 (K = 1)

    $$ \begin{aligned} A &= \left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0.450747108014309 & 0 & 0 & 0 \\ 0.176334489487884 & 0.489006055228990 & 0 &0 \\ 0.205978444248877 & 0.153461952023751 & 0.371692342116013 & 0 \end{array} \right], \\ b &= \left[ \begin{array}{cccc} 0.232713406461542 & 0.138692626440902 & 0.335920229077949 & 0.292673738019606 \end{array} \right]^{T}, \\ \tilde A &= \left[ \begin{array}{cccc} 0& 0 &0 & 0 \\ 0.278199085980253 & 0.172548022034056 & 0 & 0 \\ 0.311223289933055 & 0.067501647574067 & 0.286615607209751 & 0 \\ 0.245782336194099 & 0.248157968208214 & 0.098290863121270 & 0.138901570865059 \end{array} \right], \\ \tilde b &= \left[ \begin{array}{cccc} 0.222127817205699 & 0.224274752530891 & 0.088831226489934 & 0.464766203773476 \end{array} \right]^{T}. \end{aligned} $$
  4. 4.

    Optimized SSP5444 (K = 1)

    $$ \begin{aligned} A &= \left[ \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0\\ 0.315011539134757& 0 & 0 & 0 & 0 \\ 0.092321902233778 &0.445552754571477&0 & 0 &0\\ 0.075770251330566 & 0.114390152779218& 0.390310565760801 &0 &0\\ 0.173751459170420 & 0.132589347012580& 0.196012450887862 & 0.457940007330246 & 0 \end{array} \right], \\b & = \left[ \begin{array}{ccccc} 0.133025520352077 & 0.255101129886359 &0.140292259141287& 0.286122212426944& 0.185458878193333 \end{array} \right]^{T},\\ \tilde A & = \left[ \begin{array}{ccccc} 0& 0 & 0& 0 & 0 \\ 0.196366311461557 & 0.118645227673200 & 0& 0 &0\\ 0.256502494758585 &0.034771904397118 &0.246600257649552 &0 &0\\ 0.124682224052501 &0.279820872038586 & 0.063311571463197 & 0.112656302316301 & 0 \\ 0.105880545237959 &0.146474778500675&0.329779897287268&0.361855954830076 & 0.016302088545130 \end{array} \right], \\ \tilde b & = b. \end{aligned} $$

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Zhang, H., Yan, J., Qian, X. et al. 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 Algor 88, 1309–1336 (2021). https://doi.org/10.1007/s11075-021-01077-x

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