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Numerical Blow-Up of Nonlinear Parabolic Integro-Differential Equations on Unbounded Domain

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Abstract

The efficient numerical methods of the nonlinear parabolic integro-differential PDEs on unbounded spatial domains whose solutions blow up in finite time are considered. Based on the unified approach proposed in Zhang et al. (Phys Rev E 78:026709, 2008), Zhang et al. (Phys Rev E 79:046711, 2009), the nonlinear absorbing boundary conditions for one-dimensional and two-dimensional nonlinear parabolic integro-differential PDEs are derived. Thus the original problem on the unbounded spatial domain is reduced to an initial-boundary-value (IBV) problem on a bounded computational domain. Secondly, a simple but efficient adaptive time-stepping scheme for the reduced IBV problem is achieved by using the fixed point method to make the finite difference approximation stable at each time level. At each time level, we also prove that the lower bound and upper bound of the blow-up time can be bounded by the numerical blow-up times of the forward and backward Euler schemes. Finally, the theoretical results are illustrated by a broad range of numerical examples, including a problem with a circle line blow-up.

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Acknowledgments

HB and TT are supported by Hong Kong Research Council GRF Grants and Hong Kong Baptist University FRG grants. JZ is partially supported by the start-up fund of Beijing CSRC and NSFC 91430216, and thanks Dr. Zhan-wen Yang for valuable discussions. The authors thank referees for their carefully reading of the original version of the paper and for their valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Hermann Brunner

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Hermann Brunner

  3. Department of Mathematics & Institute for Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Tao Tang

  4. Beijing Computational Science Research Center, No. 10 Dongbeiwang West Road, Haidian District, Beijing, 100094, China

    Jiwei Zhang

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  1. Hermann Brunner
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  2. Tao Tang
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  3. Jiwei Zhang
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Correspondence to Jiwei Zhang.

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Brunner, H., Tang, T. & Zhang, J. Numerical Blow-Up of Nonlinear Parabolic Integro-Differential Equations on Unbounded Domain. J Sci Comput 68, 1281–1298 (2016). https://doi.org/10.1007/s10915-016-0179-8

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  • DOI: https://doi.org/10.1007/s10915-016-0179-8

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