Numerical methods for transport problems on networks

本研究的目的是发展有效的数值方法来模拟网络系统的双曲平衡律。在这样的网络中，每个边都是一个准一维域，通过其两端的结点与系统的其余部分相互作用。这些相互作用的特征取决于手头的应用程序;理想情况下，对它们进行建模以减轻降维的影响。在数学上，交叉点的存在使正确解的选择过程复杂化。在目前的情况下，现有的数值方法的天真使用可能是低效的，不稳定的，并导致非物理的解决方案。专门针对网络问题优化的数值方法将被设计，分析和实施。 这不仅涉及离散化问题，而且更重要的是，新的求解器的建设。这些解算器将根据微分代数方程数值方法和区域分解方法的最新进展进行设计。一些现象在大多数计算域中基本上是一维的，并且仅是“局部多维的”。 能够可靠地切换到一维近似表示显著的节省;如何有效地做到这一点将被调查。树木中的运输现象，在许多生物体中起着重要作用（呼吸，血液循环等），导致其他类型的耦合，新的数值方法也被提出。两个应用程序被认为是测试床的各个方面的研究。它们分别涉及动脉中的血液流动和气体流动。道路、管道或动脉网络在我们生活的许多方面发挥着基本作用。它们允许例如汽车、未经处理的污水、气体或血液在相应的城市、国家、生物体等中的有效运输和分配.相关的实际问题从商业（天然气管道网络的优化）和公共安全（特定地理区域的紧急疏散计划）到健康（基于患者脉管系统的中风可能性）。 虽然科学计算的工具已经非常成功地应用于许多类型的传输现象，如空气动力学问题，网络上的传输的数值模拟面临着一些尚未解决的具体挑战。将研究三个主要问题。(i)效率：这些方法必须足够灵活，以允许模拟整个网络，而不是仅模拟其中的某些部分。(ii)准确性：在交叉口或十字路口附近的流量比远离它们的流量更大。在同一网络的不同位置可能必须使用不同的模型。该项目将研究如何有效地实现这种网络流的多物理模型。(iii)最后，该项目的各种理论和数值方面将在两个具体的应用中进行测试，动脉血流和刚性管道中的气体流动。

The objective of this research is to develop efficient numerical methods for the simulation of networked systems of hyperbolic balance laws. In such networks, each edge is a quasi one-dimensional domain interacting with the rest of the system through junctions at each of its ends. The character of those interactions depends on the applications at hand; ideally, they aremodeled to mitigate the effects of dimension reduction. Mathematically, the presence of junctions complicates the selection process of proper solutions. The naive use of existing numerical methods in the present context may be inefficient, unstable and lead to nonphysical solutions. Numerical methods specifically optimized for network problems will be designed, analyzed and implemented. This involves not only discretization issues but also and more importantly the construction of new solvers. Those solvers will be designed by building on recent progress in both numerical methods for differential algebraic equations and in domain decomposition methods. Some phenomena are essentially one-dimensional in most of the computational domain and only "locally multidimensional". Being able to reliably switch to one-dimensional approximations represents significant savings; how to do this efficiently will be investigated. Transport phenomena in trees, which play an essential role in many organisms (breathing, blood circulation,etc...), lead to other types of couplings for which new numerical approaches are also proposed. Two applications are considered as test beds for various aspects of the research. They respectively involve blood flows in arteries and gas flows.Networks of roads, pipelines or arteries play a fundamental role in many aspects of our lives. They allow the efficient transport and distribution of, for instance, cars, raw sewage, gas or blood in respectively cities, countries, organisms, etc... Related practical problems range from business (optimization of natural gas pipeline networks) and public safety (emergency evacuation schedules in specific geographic areas) to health (likelihood of stroke based on patients' vasculature). While the tools of scientific computing have been applied very successfully to many types of transport phenomena such as problems in aerodynamics, the numerical simulation of transport on networks faces several specific challenges that have yet to resolved. Three main issues will be studied. (i) Efficiency: the methods have to be nimble enough to allow the simulation of entire networks as opposed to only some of their parts. (ii) Accuracy: flows are more involved near junctions or crossroads than they are away from them. Different models may have to be used at different locations of a same network. The project will study efficient implementation of such multi-physics models for network flows. (iii) Finally, the various theoretical and numerical aspects of the project will be testedon two specific applications, arterial blood flow and gas flow in rigid pipes.