A Computational Study of Residual KPP Front Speeds in Time-Periodic Cellular Flows in the Small Diffusion Limit

Abstract

The minimal speeds ($c^*$) of the Kolmogorov-Petrovsky-Piskunov (KPP) fronts at small diffusion ($ε\ll 1$) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of $c^*$ reduces the computation to that of a principal eigenvalue problem on a periodic domain of a linear advection-diffusion operator with space-time periodic coefficients and small diffusion. To solve the advection dominated time-dependent eigenvalue problem efficiently over large time, a combination of finite element and spectral methods, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling $c^*=\mathcal{O}(ε^{1/4})$ in steady cellular flows, a new relation $c^* = \mathcal{O}(1)$ as $ε\ll 1$ is revealed in the time-periodic cellular flows due to the presence of chaotic streamlines. Residual propagation speed emerges from the Lagrangian chaos which is quantified as a sub-diffusion process.