document: A multi-physics description of a multiscale system is often referred to as a ‘hybrid’ model. In fluid dynamics, a typical hybrid combines a molecular treatment (a ‘micro’ model) with a continuum-fluid one (a ‘macro’ model), with the aim of obtaining the accuracy of the former with the efficiency of the latter [1–4]. The micro and macro models generally have characteristic timescales that are very different, which means that time-accurate simulations can be extremely challenging: the size of the timestep required to make the micro model stable and accurate is so small that simulations over significant macro-scale time periods are intractable. If the system is ‘scale-separated’, a physical (as distinct from numerical) approximation can be made that enables the coupled models to advance at different rates (asynchronously) with negligible penalty on macro-scale accuracy. E et al. [5] were the first to introduce and implement this concept in a time-stepping method for coupled systems, referred to in the classification of Lockerby et al. [6] as a continuous asynchronous (CA) scheme (‘continuous’ since the micro and macro models advance without interruption [5]). In this paper we extend this idea to multiscale systems comprising an arbitrary number of coupled models.
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