Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics
For researchers in reaction-diffusion modeling, this provides a hybrid method to balance accuracy and computational cost, though the improvement over existing methods is incremental.
The paper presents two PDE-assisted Brownian dynamics (PBD) algorithms that couple particle tracking with mean-field PDEs, enabling exact simulation in parts of the domain while using a continuum description elsewhere. The overlap region variant is shown to be necessary for accurate variance computation.
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.