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Finite Element Computation of KPP Front Speeds in 3D Cellularand ABC Flows

Published online by Cambridge University Press:  12 June 2013

L. Shen ,
J. Xin  and
A. Zhou
L. Shen
Affiliation:
School of Mathematics, Capital Normal University, Beijing 100048, China
J. Xin*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
A. Zhou
Affiliation:
LSEC,Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

We carried out a computational study of propagation speeds ofreaction-diffusion-advection fronts in three dimensional (3D) cellular andArnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP)nonlinearity. The variational principle of front speeds reduces the problem to a principaleigenvalue calculation. An adaptive streamline diffusion finite element method is used inthe advection dominated regime. Numerical results showed that the front speeds areenhanced in cellular flows according to sublinear power lawO(δp),p ≈ 0.13, δ the flow intensity. In ABC flows however,the enhancement is O(δ) which can be attributed to thepresence of principal vortex tubes in the streamlines. Poincaré sections are used tovisualize and quantify the chaotic fractions of ABC flows in the phase space. The effectof chaotic streamlines of ABC flows on front speeds is studied by varying the threeparameters (a,b,c) of the ABC flows. Speed enhancement alongx direction is reduced as b (the parameter controlingthe flow variation along x) increases at fixed(a,c) > 0, more rapidly as the corresponding ABC streamlines becomemore chaotic.

Keywords

KPP fronts3D Cellular and ABC Flowseigenvalue problemsadaptive streamline diffusion finite element method
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 182 - 197
Copyright
© EDP Sciences, 2013

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