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Reaction-diffusion systems are used to represent many biological and physicalphenomena. They model the random motion of particles (diffusion) andinteractions between them (reactions). Such systems can be modelled at multiplescales with varying degrees of accuracy and computational efficiency. Whenrepresenting genuinely multiscale phenomena, fine-scale models can beprohibitively expensive, whereas coarser models, although cheaper, often lacksufficient detail to accurately represent the phenomenon at hand. Spatialhybrid methods couple two or more of these representations in order to improveefficiency without compromising accuracy.  In this paper, we present a novel spatial hybrid method, which we call theauxiliary region method (ARM), which couples PDE and Brownian-basedrepresentations of reaction-diffusion systems. Numerical PDE solutions on oneside of an interface are coupled to Brownian-based dynamics on the other sideusing compartment-based "auxiliary regions". We demonstrate that the hybridmethod is able to simulate reaction-diffusion dynamics for a number ofdifferent test problems with high accuracy. Further, we undertake erroranalysis on the ARM which demonstrates that it is robust to changes in the freeparameters in the model, where previous coupling algorithms are not. Inparticular, we envisage that the method will be applicable for a wide range ofspatial multi-scales problems including, filopodial dynamics, intracellularsignalling, embryogenesis and travelling wave phenomena.

The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems