A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit

• We find the subdiffusion from the mean square displacement of the streamline.
• We find the asymptotic solution c∗(ϵ)=O(1)>0c∗(ϵ)=O(1)>0 as ϵ↓0ϵ↓0.
• We show the large and circular layer in KPP equation.

The minimal speeds (c∗c∗) of the Kolmogorov–Petrovsky–Piskunov (KPP) fronts at small diffusion (ϵ≪1ϵ≪1) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of c∗c∗ reduces the computation to that of a principle 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 spectral methods and finite element, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling c∗=O(ϵ1/4)c∗=O(ϵ1/4) in steady cellular flows, a new relation c∗=O(1)c∗=O(1) as ϵ≪1ϵ≪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.