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

Multiscale Reaction-Diffusion Algorithms: PDE-Assisted Brownian Dynamics