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

Boyi Hu, Zhongjian Wang, Jack Xin, Zhiwen Zhang

TL;DR

The paper introduces the stochastic genetic interacting particle (SGIP) method to solve reaction-diffusion-advection (RDA) equations by combining a particle-based advection-diffusion step with a density-based reaction step, linked through adaptive resampling. It provides a rigorous convergence analysis, establishing an $L^2$ global error bound that scales with the time step, particle count, and spatial bin width under regularity assumptions. Numerical experiments in 1D, 2D, and 3D (including FKPP, cubic, and Arrhenius reactions in complex flows) show strong agreement with finite-difference solutions and demonstrate substantial computational advantages in high dimensions and advection-dominated regimes. The SGIP framework offers a scalable, mesh-free alternative for simulating multi-dimensional RDA systems and lays groundwork for extensions to multi-species problems and adaptive acceleration strategies.

Abstract

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

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

TL;DR

The paper introduces the stochastic genetic interacting particle (SGIP) method to solve reaction-diffusion-advection (RDA) equations by combining a particle-based advection-diffusion step with a density-based reaction step, linked through adaptive resampling. It provides a rigorous convergence analysis, establishing an global error bound that scales with the time step, particle count, and spatial bin width under regularity assumptions. Numerical experiments in 1D, 2D, and 3D (including FKPP, cubic, and Arrhenius reactions in complex flows) show strong agreement with finite-difference solutions and demonstrate substantial computational advantages in high dimensions and advection-dominated regimes. The SGIP framework offers a scalable, mesh-free alternative for simulating multi-dimensional RDA systems and lays groundwork for extensions to multi-species problems and adaptive acceleration strategies.

Abstract

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

Paper Structure

This paper contains 18 sections, 6 theorems, 71 equations, 14 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. SGIP Algorithm
  3. Initialization
  4. Advection-Diffusion Step
  5. Reaction Step
  6. Density Estimation
  7. Reaction Integration
  8. Closed-form Solutions
  9. General Reaction Terms
  10. Resampling Step
  11. Error Analysis of the SGIP method for RDA Equation
  12. Numerical Experiments
  13. 1D Numerical Experiments
  14. 2D Numerical Experiments
  15. 2D steady flow
  16. ...and 3 more sections

Key Result

Theorem 3

Under Assumptions ass:regularity and ass:CFL, for $n\Delta t \leq T$, the global error of the SGIP algorithm satisfies: where $C_T$ depends on final time $T$, diffusion parameter $D$, and solution regularity, but is independent of time step$\Delta t$, bin size $\Delta x$, and the number of particles $N$.

Figures (14)

  • Figure 1: Comparison of FDM and SGIP method at $u_t = u_{xx} + u(1-u)$
  • Figure 2: Comparison of FDM and SGIP method at $u_t = u_{xx} + u^2(1-u)$
  • Figure 3: Comparison of FDM and SGIP method at $u_t = u_{xx} + e^{-1/2u}(1-u)$
  • Figure 4: Propagation speed comparison for particle and FDM
  • Figure 5: Comparison of FDM and SGIP method at $u_t = u_{xx} + u_{yy} + u(1-u)$
  • ...and 9 more figures

Theorems & Definitions (13)

  • Remark 1: Fast adaptive algorithm for computational efficiency
  • Theorem 3: Global error bound
  • Lemma 1: Propagation error estimate
  • proof
  • Lemma 2: Splitting error estimate
  • proof
  • Lemma 3: Density estimation error
  • proof
  • Remark 2: On the particle motion discretization error
  • Lemma 4: Reaction integration error
  • ...and 3 more