Numerical methods for transport problems on networks

本研究的目的是发展双曲平衡律网络系统仿真的有效数值方法。在这样的网络中，每条边都是一个准一维域，通过其两端的连接与系统的其余部分相互作用。这些相互作用的性质取决于手头的应用程序；理想情况下，它们被建模以减轻维数减少的影响。从数学上讲，结点的存在使正确解的选择过程复杂化。在目前的情况下，单纯地使用现有的数值方法可能是低效的、不稳定的，并且会导致非物理解。专门针对网络问题优化的数值方法将被设计、分析和实现。这不仅涉及离散化问题，而且更重要的是涉及新的求解器的构造。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来设计。有些现象在大多数计算域中本质上是一维的，只有“局部多维”。能够可靠地切换到一维近似表示显著节省；如何有效地做到这一点将被研究。树木中的传输现象在许多生物体（呼吸、血液循环等）中起着至关重要的作用，导致了其他类型的耦合，也提出了新的数值方法。两个应用程序被认为是研究的各个方面的试验台。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重要作用。它们允许汽车、未经处理的污水、天然气或血液在各自的城市、国家、生物等进行有效的运输和分配。相关的实际问题范围从商业（天然气管道网络的优化）、公共安全（特定地理区域的紧急疏散时间表）到健康（基于患者血管系统的中风可能性）。虽然科学计算工具已经非常成功地应用于许多类型的输运现象，如空气动力学问题，但网络输运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：方法必须足够灵活，能够模拟整个网络，而不是只模拟其中的某些部分。（ii）准确性：在路口或十字路口附近的流量比远离路口或十字路口的流量更大。不同的模型可能必须在同一网络的不同位置使用。该项目将研究网络流的这种多物理场模型的有效实现。（iii）最后，项目的各种理论和数值方面将在两个具体应用中进行测试，即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.