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

TL;DR

This study investigates how small diffusion affects the minimal propagation speeds of KPP fronts in time-periodic cellular flows with chaotic streamlines, revealing a new constant order relation due to chaos.

Contribution

It introduces a novel relation showing residual front speeds are order one in time-periodic flows, contrasting with steady flows, and develops efficient numerical methods for eigenvalue problems.

Findings

Residual propagation speed is order one in time-periodic flows.

Chaotic streamlines induce sub-diffusive behavior affecting front speeds.

New scaling relation $c^* = ext{O}(1)$ for small diffusion in time-periodic flows.

Abstract

The minimal speeds ($c^*$) of the Kolmogorov-Petrovsky-Piskunov (KPP) fronts at small diffusion ($\epsilon \ll 1$) 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 $c^*=\mathcal{O}(\epsilon^{1/4})$ in steady cellular flows, a new relation $c^* = \mathcal{O}(1)$ as $\epsilon \ll 1$ 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.