README 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 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.诸多生物与物理系统会在多空间尺度、多时间尺度或多群体尺度下展现出不同的行为特性。利用数值技术对多尺度过程进行模拟时，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，通常具备较快的模拟速度，但在低浓度区域或小空间尺度区域中，它们无法提供所需的个体层面细节信息。然而，若要在整个计算域内以及高浓度区域都采用个体层面的精细模拟，其计算成本会十分高昂。空间耦合混合方法则搭建起了桥梁：通过将空间划分为不同的建模子域，该类方法能够在同一个空间域内对同一物种采用多种表征方式。在过去二十年间，这类混合方法逐渐得到学界广泛关注，如今已成为化学、物理学、数学等多个学科领域中极具活力的研究方向。本次综述主要出于三大目的：其一，我们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章中，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找到适配自身需求的方案；其二，我们提供了包含算法与配套代码的标准示例，用以展示这类方法在实际场景中的运作逻辑；其三，我们梳理了将这类方法应用于真实生物与物理问题的相关研究文献，以此验证该类方法的实用价值。此外，我们还探讨了混合方法开发领域中的若干开放性研究问题，并对该领域的未来发展方向进行了展望。