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

Liyao Lyu, Huan Lei

TL;DR

The paper introduces a stochastic branching particle method for nonlinear non-conservative advection–diffusion–reaction equations, leveraging a two-step operator splitting into advection–diffusion and reaction components. By mapping the PDE to a measure-valued, particle-based framework and employing a birth–death branching mechanism, it achieves a mesh-free, nonnegativity-preserving discretization that remains robust in the presence of singularities and blow-up. Convergence analysis for the projected scalar problem is provided, with error bounds that separate projection error and Monte Carlo variance, and the method is extended to the Keller–Segel system with two coupled fields. Numerical experiments on Allen-Cahn and Keller-Segel demonstrate accurate nonlinear dynamics and 1/√N convergence, illustrating the approach’s potential for high-dimensional and complex non-conservative PDEs without adaptive meshing.

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.

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

TL;DR

The paper introduces a stochastic branching particle method for nonlinear non-conservative advection–diffusion–reaction equations, leveraging a two-step operator splitting into advection–diffusion and reaction components. By mapping the PDE to a measure-valued, particle-based framework and employing a birth–death branching mechanism, it achieves a mesh-free, nonnegativity-preserving discretization that remains robust in the presence of singularities and blow-up. Convergence analysis for the projected scalar problem is provided, with error bounds that separate projection error and Monte Carlo variance, and the method is extended to the Keller–Segel system with two coupled fields. Numerical experiments on Allen-Cahn and Keller-Segel demonstrate accurate nonlinear dynamics and 1/√N convergence, illustrating the approach’s potential for high-dimensional and complex non-conservative PDEs without adaptive meshing.

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.

Paper Structure

This paper contains 11 sections, 3 theorems, 78 equations, 9 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Stochastic branching method
  3. Initial condition and Normalization
  4. Operator Splitting
  5. Advection Diffusion Step
  6. Branching Step
  7. Nonlinear Interpolation
  8. Error Estimations
  9. Extension to PDE system
  10. Numerical Examples
  11. Discussion

Key Result

Proposition 1

Let $S=\{\mathbf{X}_i\}_{i\in \mathcal{I}}$ be a set of particles, whose empirical measure is defined as $\mu (\mathrm{d} \mathbf{x}) = \frac{1}{N}\sum_{i\in \mathcal{I} }\delta_{\mathbf{X}_i}$. If the particles are updated according to the above rules then the updated particle set $S_+=\{ \mathbf{X for all $f\in C^2_b(\mathbb R)$.

Figures (9)

  • Figure 1: Numerical solution of the AC equation \ref{['eq:ac_pde']} at different time snapshots ($T=0, 0.2, 0.4, 0.6, 0.8$). The top row displays the reference solution computed by a finite difference method, while the bottom row shows the results from our proposed particle method.
  • Figure 2: Time evolution of the total mass of $v$ for different numbers of particles. The left panel shows the total mass over time, while the right panel displays the corresponding error in the total mass.
  • Figure 3: Comparison of the numerical approximation of $u$ obtained from the finite difference method with a $100 \times 100$ grid and the present particle-based method at $t = 0.0, 0.1, 0.2, 0.3, 0.4$.
  • Figure 4: Comparison of the numerical approximation of $v$ obtained from the finite difference method with a $100 \times 100$ grid and the present particle-based method at $t = 0.0, 0.1, 0.2, 0.3, 0.4$.
  • Figure 5: Relative $L^2$ error of $u$ and $v$ with respect to the number of samples $N_0$. The reference solution is obtained with $N_0 = 320,000$.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1