Article
| Title: | Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: $L^2$ stability (English) |
| Author: | Vacek, Lukáš |
| Author: | Shu, Chi-Wang |
| Author: | Kučera, Václav |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 70 |
| Issue: | 3 |
| Year: | 2025 |
| Pages: | 311-339 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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. (English) |
| Keyword: | traffic flow |
| Keyword: | discontinuous Galerkin method |
| Keyword: | Godunov numerical flux |
| Keyword: | $L^2$ stability |
| MSC: | 35L04 |
| MSC: | 65M60 |
| MSC: | 76a30 |
| DOI: | 10.21136/AM.2025.0017-25 |
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| Date available: | 2025-07-01T12:17:51Z |
| Last updated: | 2025-07-07 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153021 |
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| Reference: | [11] Vacek, L., Kučera, V.: Discontinuous Galerkin method for macroscopic traffic flow models on networks.Commun. Appl. Math. Comput. 4 (2022), 986-1010. Zbl 1513.65383, MR 4446828, 10.1007/s42967-021-00169-8 |
| Reference: | [12] Vacek, L., Kučera, V.: Godunov-like numerical fluxes for conservation laws on networks.J. Sci. Comput. 97 (2023), Article ID 70, 27 pages. Zbl 1526.65047, MR 4663639, 10.1007/s10915-023-02386-0 |
| Reference: | [13] Vacek, L., Shu, C.-W., Kučera, V.: Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle.(to appear) in Appl. Math., Praha (2025). MR 4816401, 10.21136/AM.2025.0018-25 |
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