Article
| Title: | Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle (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: | 341-366 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We prove the maximum principle for a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks described by the Lighthill-Whitham-Richards equations. The paper is a followup of the preceding paper, Part I, where $L^2$ stability of the scheme is analyzed. At traffic junctions, we consider numerical fluxes based on Godunov's flux derived in our previous work. We also construct a new Godunov-like numerical flux taking into account right of way at the junction to cover a wider variety of scenarios in the analysis. These fluxes are easily constructible for any number of incoming and outgoing roads, respecting the drivers' preferences. We prove that the explicit Euler or SSP DG scheme with limiters satisfies a maximum principle on general networks. Numerical experiments demonstrate the obtained results. (English) |
| Keyword: | traffic flow |
| Keyword: | discontinuous Galerkin method |
| Keyword: | Godunov numerical flux |
| Keyword: | maximum principle |
| MSC: | 35L04 |
| MSC: | 65M60 |
| MSC: | 76a30 |
| DOI: | 10.21136/AM.2025.0018-25 |
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| Date available: | 2025-07-01T12:18:21Z |
| Last updated: | 2025-07-07 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153023 |
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| Reference: | [7] 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: | [8] 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: | [9] Vacek, L., Shu, C.-W., Kučera, V.: Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: $L^2$ stability.(to appear) in Appl. Math., Praha (2025). MR 4816401, 10.21136/AM.2025.0017-25 |
| Reference: | [10] Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws.J. Comput. Phys. 229 (2010), 3091-3120. Zbl 1187.65096, MR 2601091, 10.1016/j.jcp.2009.12.030 |
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