Applications of Mathematics, first online, pp. 1-29
Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: $L^2$ stability
Lukáš Vacek, Chi-Wang Shu, Václav Kučera
Received January 17, 2025. Published online June 12, 2025.
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 $L^2$-norm. The resulting estimates allow for a linear-in-time growth of the square of the $L^2$-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.
References: [1] S. Canic, B. Piccoli, J.-M. Qiu, T. Ren: Runge-Kutta discontinuous Galerkin method for traffic flow model on networks. J. Sci. Comput. 63 (2015), 233-255. DOI 10.1007/s10915-014-9896-z | MR 3315275 | Zbl 1321.90034
[2] B. Cockburn, C.-W. Shu: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: General framework. Math. Comput. 52 (1989), 411-435. DOI 10.2307/2008474 | MR 0983311 | Zbl 0662.65083
[3] V. Dolejší, M. Feistauer: Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48. Springer, Cham (2015). DOI 10.1007/978-3-319-19267-3 | MR 3363720 | Zbl 1401.76003
[4] M. Garavello, B. Piccoli: Traffic Flow on Networks. AIMS Series on Applied Mathematics 1. American Institute of Mathematical Sciences, Springfield (2006). MR 2328174 | Zbl 1136.90012
[5] B. D. Greenshields: A study of traffic capacity. Proceedings of the Fourteenth Annual Meeting of the Highway Research Board Held at Washington, D.C. December 6-7, 1934. Part I. Highway Research Board, Kansas (1935), 448-477.
[6] G. Jiang, C.-W. Shu: On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62 (1994), 531-538. DOI 10.1090/S0025-5718-1994-1223232-7 | MR 1223232 | Zbl 0801.65098
[7] A. Jüngel: Modeling and numerical approximation of traffic flow problems. Available at https://www.tuwien.at/index.php?eID=dumpFile&t=f&f=188859&token=cafe6d48521f4797d67404fd0e0e07cbe8f06cbc (2002), 1-32.
[8] P. Kachroo, S. Sastry: Traffic Flow Theory: Mathematical Framework. University of California Berkeley, Berkeley (2012).
[9] W. H. Reed, T. R. Hill: Triangular mesh methods for the neutron transport equation. National Topical Meeting on Mathematical Models and Computational Techniques for Analysis of Nuclear Systems, Ann Arbor, Michigan, USA, 8 Apr 1973. Los Alamos Scientific Laboratory, Los Alamos (1973), 23 pages.
[10] C.-W. Shu: Discontinuous Galerkin methods: General approach and stability. Available at https://www3.nd.edu/~zxu2/acms60790S15/DG-general-approach.pdf (2009), 44 pages.
[11] L. Vacek, V. Kučera: Discontinuous Galerkin method for macroscopic traffic flow models on networks. Commun. Appl. Math. Comput. 4 (2022), 986-1010. DOI 10.1007/s42967-021-00169-8 | MR 4446828 | Zbl 1513.65383
[12] L. Vacek, V. Kučera: Godunov-like numerical fluxes for conservation laws on networks. J. Sci. Comput. 97 (2023), Article ID 70, 27 pages. DOI 10.1007/s10915-023-02386-0 | MR 4663639 | Zbl 1526.65047
[13] L. Vacek, C.-W. Shu, V. Kučera: Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle. To appear in Appl. Math., Praha (2025). DOI 10.21136/AM.2025.0018-25
[14] F. van Wageningen-Kessels, H. van Lint, K. Vuik, S. Hoogendoorn: Genealogy of traffic flow models. EURO J. Transport. Log. 4 (2015), 445-473. DOI 10.1007/s13676-014-0045-5
Affiliations: Lukáš Vacek (corresponding author), Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: lvacek@karlin.mff.cuni.cz; Chi-Wang Shu, Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA, e-mail: chi-wang_shu@brown.edu; Václav Kučera, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: kucera@karlin.mff.cuni.cz
