A stochastic branching particle method for solving non-conservative reaction-diffusion equations

TL;DR

This paper introduces a stochastic branching particle method for efficiently solving nonlinear non-conservative reaction-diffusion equations, capable of handling singularities and complex behaviors without mesh refinement.

Contribution

The paper presents a novel mesh-free stochastic particle scheme combining advection-diffusion and branching processes for non-conservative systems.

Findings

Accurately captures phase separation in Allen-Cahn equation

Effectively models chemotaxis aggregation in Keller-Segel system

Remains robust near singularities or blow-up scenarios

Abstract

We propose a stochastic branching particle-based method for solving nonlinear non-conservative advection-diffusion-reaction equations. The method splits the evolution into an advection-diffusion step, based on a linearized Kolmogorov forward equation and approximated by stochastic particle transport, and a reaction step implemented through a branching birth-death process that provides a consistent temporal discretization of the underlying reaction dynamics. This construction yields a mesh-free, nonnegativity-preserving scheme that naturally accommodates non-conservative systems and remains robust in the presence of singularities or blow-up. We validate the method on two representative two-dimensional systems: the Allen-Cahn equation and the Keller-Segel chemotaxis model. In both cases, the present method accurately captures nonlinear behaviors such as phase separation and aggregation, and achieves reliable performance without the need for adaptive mesh refinement.