A convergent interacting particle method and computation of KPP front speeds in chaotic flows

TL;DR

This paper introduces an efficient Lagrangian particle method to compute KPP front speeds in complex, chaotic, and time-periodic flows, with rigorous convergence analysis and numerical validation.

Contribution

It develops a novel particle-based numerical approach for eigenvalue problems related to reaction-diffusion-advection fronts in chaotic flows, including convergence guarantees.

Findings

Accurate computation of KPP front speeds in chaotic flows

Efficient numerical methods validated on 3D flows

Demonstrated convergence and effectiveness of the proposed approach

Abstract

In this paper, we study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficients on a periodic domain. To this end, we develop efficient Lagrangian particle 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.