GCM 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 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在多重空间、时间或种群尺度下展现出各异的动力学行为。多尺度过程在借助数值技术开展模拟时往往面临诸多挑战。尽管诸如偏微分方程（partial differential equations）这类粗粒度数值方法通常具备较快的模拟效率，但在低浓度区域或小空间尺度场景中，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乃至高浓度区域均采用个体层面的模拟方式，其计算成本将极为高昂，难以承受。空间耦合混合方法（spatially coupled hybrid methods）提供了一种可行的折中方案：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方式。在过去二十年中，这类混合方法逐渐获得广泛认可，如今已成为化学、物理学、数学等多个学科领域内极具活力的研究方向。撰写本篇综述主要出于三大核心动因：其一，我们整理了大量空间扩展型混合方法（spatially extended hybrid methods），将其整合至一篇逻辑连贯的综述中，并对各类方法开展对比分析，以便需要使用多尺度混合方法的研究者能够筛选出最贴合自身需求的最优方案；其二，我们提供了附带算法与配套源代码的标准示例（canonical examples），用以直观展示这类方法在实际场景中的运作机制；其三，我们梳理了将这类方法应用于实际生物与物理问题的相关研究文献，以此证明其实际应用价值与有效性。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。