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

¹ School of Mathematics, Capital Normal University, Beijing 100048, China
² Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
³ 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: jxin@math.uci.edu

Abstract

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