Computing KPP front speeds in time-periodic cellular and chaotic flows using an efficient Lagrangian method

In this paper, we study propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. The variational principle reduces the computation of KPP front speeds to a principal eigenvalue problem on a periodic domain of a linear advection-diffusion operator with space-time periodic coefficients. We develop efficient Lagrangian methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting methods for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical methods. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space. AMS subject classification: 35K57, 47D08, 65C35, 65L20, 65N25.