A Convergent Interacting Particle Method for Computing KPP Front Speeds in Random Flows

Abstract

<p>This paper aims to efficiently compute the spreading speeds of reaction-diffusion-advection fronts in divergence-free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We develop 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 formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method, which is implemented as a genetic evolution algorithm. The particles undergo advection-diffusion and mutation/selection through a fitness function that originates in the Feynman-Kac semigroup. We analyze the convergence of the algorithm based on operator splitting and 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 easy 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.</p>