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Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations

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Abstract

In this paper, an explicit numerical method and its fast implementation are proposed and discussed for the solution of a wide class of semilinear parabolic equations including the Allen–Cahn equation as a special case. The method combines decompositions of compact spatial difference operators on a regular mesh with stable and accurate exponential time integrators and efficient discrete FFT-based algorithms. It can deal with stiff nonlinearity and both homogeneous and inhomogeneous boundary conditions of different types based on multistep approximations and analytic evaluations of time integrals. Numerical experiments demonstrate effectiveness of the new method for both linear and nonlinear model problems.

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Author information

Authors and Affiliations

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

    Lili Ju

  2. Computer Network Information Center, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Jian Zhang

  3. LMIB and School of Mathematics and Systems Science, BeiHang University, Beijing , 100191, People’s Republic of China

    Liyong Zhu

  4. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Qiang Du

Authors
  1. Lili Ju
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  2. Jian Zhang
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  3. Liyong Zhu
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  4. Qiang Du
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Corresponding author

Correspondence to Lili Ju.

Additional information

L. Ju’s research is partially supported by the US National Science Foundation under Grant Number DMS-1215659 and the U.S. Department of Energy under Grant Number DE-SC0008087-ER65393. J. Zhang’s research is partially supported by the Natural Science Foundation of China under Grant Numbers 11271350 and 91130019. L. Zhu’s research is partially supported by the Natural Science Foundation of China under Grant Number 91130019, ISTCP of China under Grant Number 2010DFR00700, China Fundamental Research of Civil Aircraft under Grant Number MJ-F-2012-04, and the State Key Laboratory of Software Development Environment under Grant Number SKLSDE-2014ZX-03. Q. Du’s research is partially supported by the US National Science Foundation under Grant Number DMS-1318586.

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Ju, L., Zhang, J., Zhu, L. et al. Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations. J Sci Comput 62, 431–455 (2015). https://doi.org/10.1007/s10915-014-9862-9

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  • DOI: https://doi.org/10.1007/s10915-014-9862-9

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Mathematics Subject Classification (2000)