Abstract
Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as partial differential equations are typically fast to simulate, they lack the individual-level detail that may be required in regions of low concentration or small spatial scale. However, to simulate at such an individual level throughout a domain and in regions where concentrations are high can be computationally expensive. Spatially coupled hybrid methods provide a bridge, allowing for multiple representations of the same species in one spatial domain by partitioning space into distinct modelling subdomains. Over the past 20 years, such hybrid methods have risen to prominence, leading to what is now a very active research area across multiple disciplines including chemistry, physics and mathematics. There are three main motivations for undertaking this review. Firstly, we have collated a large number of spatially extended hybrid methods and presented them in a single coherent document, while comparing and contrasting them, so that anyone who requires a multiscale hybrid method will be able to find the most appropriate one for their need. Secondly, we have provided canonical examples with algorithms and accompanying code, serving to demonstrate how these types of methods work in practice. Finally, we have presented papers that employ these methods on real biological and physical problems, demonstrating their utility. We also consider some open research questions in the area of hybrid method development and the future directions for the field.
Highlights
The requirement for multiscale models arises naturally from many biological and physical scenarios due to their inherent complexity
We have explored the rich and diverse field of spatial hybrid methods, and illustrated how they can be used in order to probe previously intractable problems in the biological and physical sciences
Even for a single phenomenon, populations can vary over orders of magnitude, making traditional modelling approaches difficult
Summary
The requirement for multiscale models arises naturally from many biological and physical scenarios due to their inherent complexity. Coupled hybrid methods of the sort we cover in this review rely on the assumption that different regions of the spatial domain can be accurately represented using modelling paradigms at different scales [41,42,43,44,45] The motivation for these methods will typically be either a separation in the scale of species copy numbers in distinct regions of the domain or a requirement for a detailed model on small spatial scales. Even if there is no significant difference in copy numbers throughout the domain, there may be a small region of space which requires fine-level modelling locally, but which can tolerate coarser modelling further away in regions that are not sensitive to the individual dynamics These methods are used to represent phenomena in which boundary effects are important [1].
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