A Stochastic Genetic Interacting Particle Method for Reaction-Diffusion-Advection Equations

TL;DR

This paper introduces a stochastic genetic interacting particle method (SGIP) for solving reaction-diffusion-advection equations, combining operator splitting, adaptive resampling, and particle density estimation to improve accuracy and stability in complex flow scenarios.

Contribution

The paper presents a novel SGIP method that integrates adaptive resampling with operator splitting for reaction-diffusion-advection equations, providing rigorous error analysis and demonstrating superior performance in complex flows.

Findings

Accurate solutions for various reaction types and flow configurations.

Effective long-term stability through adaptive resampling.

Comparable or better accuracy than finite difference methods with computational benefits.

Abstract

We develop and analyze a stochastic genetic interacting particle method (SGIP) for reaction-diffusion-advection (RDA) equations. The SGIP method employs operator splitting to approximate the advection-diffusion and reaction processes, treating the former using particle drift-diffusion and the latter via exact or implicit integration of reaction dynamics over bins, where particle density is estimated using a histogram. A key innovation is the incorporation of adaptive resampling to close the loop of particle and density field description of solutions, mimicking the selection mechanism in genetics. Resampling is also crucial for maintaining long-term stability by redistributing particles in accordance with the evolving density field. We provide a comprehensive error analysis and establish convergence bounds under appropriate regularity assumptions. Numerical experiments in one to three space dimensions demonstrate the method's effectiveness across various reaction types (Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP), cubic, Arrhenius) and flow configurations (shear, cellular, cat's eye, Arnold-Beltrami-Childress (ABC) flows), showing excellent agreement with the finite difference method (FDM) while offering computational advantages for complex flow geometries and higher-dimensional problems.