标题： 非局部输运占主导的偏微分方程模型的数值方法及其在生物学中的应用 显示英文标题

标题： Numerical Approaches for non-local Transport-Dominated PDE Models with Applications to Biology

摘要： 对流占主导的偏微分方程模型在过去二十年里被广泛用于描述细胞生物学和生态学中的各种集体迁移现象。为了理解这些模型（以及它们所描述的生物系统）的行为，已经使用了不同的分析方法和数值方法。尽管最近的综述研究讨论了分析方法，但数值方法仍然面临不同的开放问题，并且因此对于每一种开发的非局部模型都是以一种临时拼凑的方式被采用。 本文综述的基本目标是总结这些对流占主导的非局部模型背后的基本思想，讨论目前用于模拟这些模型的数值方法，并最终讨论一些与这些数值方法应用相关的开放问题，特别是有限元方法的应用。 这使我们能够强调这种数值方法所提供的机会，以推动该领域研究的发展。 此外，我们详细介绍了用于离散化这些非局部方程的一些数值方案；特别是我们引入的一种新的半隐式方案，用以稳定经典方案所得出的振荡。 显示英文摘要

摘要： Transport-dominated partial differential equation models have been used extensively over the past two decades to describe various collective migration phenomena in cell biology and ecology. To understand the behaviour of these models (and the biological systems they describe) different analytical and numerical approaches have been used. While the analytical approaches have been discussed by different recent review studies, the numerical approaches are still facing different open problems, and thus are being employed on a rather ad-hoc basis for each developed non-local model. The goal of this review is to summarise the basic ideas behind these transport-dominated non-local models, to discuss the current numerical approaches used to simulate these models, and finally to discuss some open problems related to the applications of these numerical methods, in particular the finite element method. This allows us to emphasize the opportunities offered by this numerical method to advance the research in this field. In addition, we present in detail some numerical schemes that we used to discretize these non-local equations; in particular a new semi-implicit scheme we introduced to stabilize the oscillations obtained with classical schemes.