Trap Rule 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、多时间尺度或多种群尺度下呈现出复杂的行为特征。针对多尺度过程开展数值模拟时，往往会面临诸多技术挑战。尽管诸如偏微分方程（partial differential equations）这类粗粒度数值方法通常模拟效率较高，但在低浓度或小空间尺度的区域中，它们无法提供所需的个体层面细节信息。然而，若要在整个模拟域及浓度较高的区域均以个体层面开展模拟，则会产生极高的计算成本。空间耦合混合方法（spatially coupled hybrid methods）则搭建了有效的桥梁：通过将空间划分为不同的建模子域，可在同一空间域内对同一建模组分实现多种表征方式。在过去20年间，这类混合方法逐渐受到学界广泛关注，如今已成为化学、物理学、数学等多学科领域中极具活力的研究方向。撰写本综述主要有三大动因：其一，我们梳理了大量空间扩展型混合方法（spatially extended hybrid methods），并将其整合至一篇逻辑连贯的综述中，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（multiscale hybrid method）的研究人员能够快速找到适配自身需求的方案；其二，我们提供了带有算法及配套代码的典型示例，用以展示这类方法在实际场景中的具体运作流程；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，并展望了该领域的未来发展方向。