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Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations

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Abstract

In this paper we propose fast high-order numerical methods for solving a class of second-order semilinear parabolic equations in regular domains. The proposed methods are explicit in nature, and use exponential time differencing and Runge–Kutta approximations in combination with a linear splitting technique to achieve accurate and stable time integration. A two-step compact difference scheme is employed for spatial discretization to obtain fourth-order accuracy and make use of FFT-based fast calculations. Such methods can be applied to problems with stiff nonlinearities and boundary conditions of Dirichlet or periodic types. Linear stability analysis and various numerical experiments are also presented to demonstrate accuracy and stability of the proposed methods.

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Authors and Affiliations

  1. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China

    Liyong Zhu

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

    Lili Ju

  3. School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

    Weidong Zhao

Authors
  1. Liyong Zhu
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  2. Lili Ju
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  3. Weidong Zhao
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Corresponding author

Correspondence to Lili Ju.

Additional information

L. Zhu’s research is partially supported by China Fundamental Research of Civil Aircraft under grant number MJ-F-2012-04. L. Ju’s research is partially supported by US National Science Foundation under grant numbers DMS-1521965 and DMS-1215659 and by US Department of Energy under grant number DE-SC0008087-ER65393. W. Zhao’s research is partially supported by National Natural Science Foundation of China under Grant Number 11171189.

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Zhu, L., Ju, L. & Zhao, W. Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations. J Sci Comput 67, 1043–1065 (2016). https://doi.org/10.1007/s10915-015-0117-1

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  • DOI: https://doi.org/10.1007/s10915-015-0117-1

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