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Finite Element Computation of KPP Front Speeds in Cellular and Cat’s Eye Flows

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Abstract

We compute the front speeds of the Kolmogorov-Petrovsky-Piskunov (KPP) reactive fronts in two prototypes of periodic incompressible flows (the cellular flows and the cat’s eye flows). The computation is based on adaptive streamline diffusion methods for the advection-diffusion type principal eigenvalue problem associated with the KPP front speeds. In the large amplitude regime, internal layers form in eigenfunctions. Besides recovering known speed growth law for the cellular flow, we found larger growth rates of front speeds in cat’s eye flows due to the presence of open channels, and the front speed slowdown due to either zero Neumann boundary condition at domain boundaries or frequency increase of cat’s eye flows.

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Acknowledgements

LS was partially supported by National Science Foundation of China under grants 10801101, 10871198 and 11171232. JX was partially supported by NSF under grants DMS-0712881, DMS-0911277 and DMS-1211179. AZ was partially supported by the National Science Foundation of China under grants 10871198 and 10971059, the Funds for Creative Research Groups of China under grant 11021101, and the National Basic Research Program of China under grant 2011CB309703. LS would like to thank Dr. Zhiqiang Sheng at Beijing Institute of Applied Physics and Computational Mathematics for providing the Stabilized BICG solver. Part of the data are computed on the supercomputer O3800 in the Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese academy of Sciences.

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Authors and Affiliations

  1. School of Mathematics, Capital Normal University, Beijing, 100048, China

    Lihua Shen

  2. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Jack Xin

  3. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Aihui Zhou

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  1. Lihua Shen
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  2. Jack Xin
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Correspondence to Lihua Shen.

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Shen, L., Xin, J. & Zhou, A. Finite Element Computation of KPP Front Speeds in Cellular and Cat’s Eye Flows. J Sci Comput 55, 455–470 (2013). https://doi.org/10.1007/s10915-012-9641-4

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  • DOI: https://doi.org/10.1007/s10915-012-9641-4

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