A convergent interacting particle method for computing KPP front speeds in random flows

TL;DR

This paper introduces a mesh-free, adaptive interacting particle method for efficiently computing the spreading speeds of reaction-diffusion fronts in complex, high-dimensional random flows, validated through numerical experiments.

Contribution

The paper develops a stochastic interacting particle method based on the FK formula for eigenvalue problems in random flows, demonstrating convergence and efficiency in high-dimensional settings.

Findings

Method accurately computes front speeds in 2D and 3D flows.

Mesh-free approach simplifies implementation in complex domains.

Efficient for high-dimensional, advection-dominated regimes.

Abstract

We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.