Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics

TL;DR

This paper introduces two novel algorithms that integrate Brownian dynamics with PDEs for efficient multiscale reaction-diffusion simulations, enabling accurate particle tracking and variance computation across different domain regions.

Contribution

The paper presents two new PDE-assisted Brownian dynamics algorithms that seamlessly combine particle-based and continuum models for reaction-diffusion systems.

Findings

Both algorithms accurately track particles near interfaces.

The overlap region improves variance calculations.

Numerical examples demonstrate effectiveness.

Abstract

Two algorithms that combine Brownian dynamics (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface which partitions the domain and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that to accurately compute variances using the PBD simulation requires the overlap region. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented.