Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: L2 stability

IF 0.7 4区 数学 Q4 MATHEMATICS, APPLIED
Lukáš Vacek, Chi-Wang Shu, Václav Kučera
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引用次数: 0

Abstract

We study the stability of a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks. We discretize the Lighthill-Whitham-Richards equations on each road by DG. At traffic junctions, we consider two types of numerical fluxes that are based on Godunov’s numerical flux derived in a previous work of ours. These fluxes are easily constructible for any number of incoming and outgoing roads, respecting the drivers’ preferences. The analysis is split into two parts: in Part I, contained in this paper, we analyze the stability of the resulting numerical scheme in the L2-norm. The resulting estimates allow for a linear-in-time growth of the square of the L2-norm of the DG solution. This is observed in numerical experiments in certain situations with traffic congestions. Next, we prove that under certain assumptions on the junction parameters (number of incoming and outgoing roads and drivers’ preferences) the DG solution satisfies an entropy inequality where the square entropy is nonincreasing in time. Numerical experiments are presented. The work is complemented by the followup paper, Part II, where a maximum principle is proved for the DG scheme with limiters.

网络上交通流的类godunov数值流的不连续伽辽金方法。第一部分:L2稳定性
研究了应用于网络交通流问题数值解的不连续伽辽金方法的稳定性。我们用DG离散每条道路上的lighhill - whitham - richards方程。在交通路口,我们考虑两种类型的数值通量,它们是基于我们在以前的工作中导出的Godunov数值通量。这些通量很容易构建任何数量的进出道路,尊重司机的喜好。本文的分析分为两部分:第一部分分析了所得到的数值格式在l2范数下的稳定性。所得到的估计允许DG解的l2范数的平方的线性增长。在某些交通拥堵情况下的数值实验中可以观察到这一点。接下来,我们证明了在对交叉口参数(进出道路数量和驾驶员偏好)的某些假设下,DG解满足熵不等式,其中平方熵随时间不增加。给出了数值实验结果。这项工作是补充了后续文件,第2部分,其中最大原则证明了DG方案与限制。
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来源期刊
Applications of Mathematics
Applications of Mathematics 数学-应用数学
CiteScore
1.50
自引率
0.00%
发文量
0
审稿时长
3.0 months
期刊介绍: Applications of Mathematics publishes original high quality research papers that are directed towards applications of mathematical methods in various branches of science and engineering. The main topics covered include: - Mechanics of Solids; - Fluid Mechanics; - Electrical Engineering; - Solutions of Differential and Integral Equations; - Mathematical Physics; - Optimization; - Probability Mathematical Statistics. The journal is of interest to a wide audience of mathematicians, scientists and engineers concerned with the development of scientific computing, mathematical statistics and applicable mathematics in general.
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