{"title":"Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: L2 stability","authors":"Lukáš Vacek, Chi-Wang Shu, Václav Kučera","doi":"10.21136/AM.2025.0017-25","DOIUrl":null,"url":null,"abstract":"<div><p>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 <i>L</i><sup>2</sup>-norm. The resulting estimates allow for a linear-in-time growth of the square of the <i>L</i><sup>2</sup>-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.</p></div>","PeriodicalId":55505,"journal":{"name":"Applications of Mathematics","volume":"70 3","pages":"311 - 339"},"PeriodicalIF":0.7000,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applications of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.21136/AM.2025.0017-25","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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.
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.