PCM CODE from Spatially extended hybrid methods: a review

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 with a need for 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.

诸多生物与物理系统在多重空间、时间乃至种群尺度上均展现出各异的行为特征。对多尺度过程采用数值技术进行模拟时，往往会面临诸多挑战。诸如偏微分方程（partial differential equations）这类粗粒度模拟方法，通常具备较快的计算速度，但无法捕捉低浓度区域或小空间尺度区域所需的个体层面细节。然而，若要在整个计算域内乃至高浓度区域均采用个体层面的模拟方式，则会产生极高的计算成本。空间耦合混合方法（spatially coupled hybrid methods）为此提供了可行的解决方案：通过将空间划分为多个独立的建模子域，可在同一空间域内对同一物种采用多种表征形式。
在过去二十年间，这类混合方法逐渐崭露头角并受到广泛关注，如今已成为化学、物理、数学等多个学科中极具活力的研究领域。
撰写本篇综述主要出于三大动因：其一，本文梳理了大量空间扩展型混合方法，并将其整合至一篇逻辑连贯的综述文本中，同时对各类方法进行对比分析，以便需要使用多尺度混合方法的研究者能够快速筛选出最适配自身需求的方法；其二，本文提供了搭载算法与配套代码的标准范例，用以直观展示这类方法的实际运作流程；其三，本文收录了将这类方法应用于实际生物与物理问题的相关研究成果，以此验证该类方法的实用价值。此外，本文还探讨了混合方法开发领域中尚未解决的若干研究问题，并对该领域的未来发展方向进行了展望。