Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle

Lukáš Vacek; Chi-Wang Shu; Václav Kučera

Applications of Mathematics (2025)

  • Issue: 3, page 341-366
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Vacek, Lukáš, Shu, Chi-Wang, and Kučera, Václav. "Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle." Applications of Mathematics (2025): 341-366. <http://eudml.org/doc/299992>.

@article{Vacek2025,
abstract = {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.},
author = {Vacek, Lukáš, Shu, Chi-Wang, Kučera, Václav},
journal = {Applications of Mathematics},
keywords = {traffic flow; discontinuous Galerkin method; Godunov numerical flux; maximum principle},
language = {eng},
number = {3},
pages = {341-366},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle},
url = {http://eudml.org/doc/299992},
year = {2025},
}

TY - JOUR
AU - Vacek, Lukáš
AU - Shu, Chi-Wang
AU - Kučera, Václav
TI - Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle
JO - Applications of Mathematics
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 3
SP - 341
EP - 366
AB - 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.
LA - eng
KW - traffic flow; discontinuous Galerkin method; Godunov numerical flux; maximum principle
UR - http://eudml.org/doc/299992
ER -

References

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  7. Vacek, L., Kučera, V., 10.1007/s42967-021-00169-8, Commun. Appl. Math. Comput. 4 (2022), 986-1010. (2022) Zbl1513.65383MR4446828DOI10.1007/s42967-021-00169-8
  8. Vacek, L., Kučera, V., 10.1007/s10915-023-02386-0, J. Sci. Comput. 97 (2023), Article ID 70, 27 pages. (2023) Zbl1526.65047MR4663639DOI10.1007/s10915-023-02386-0
  9. Vacek, L., Shu, C.-W., Kučera, V., 10.21136/AM.2025.0017-25, (to appear) in Appl. Math., Praha (2025). MR4816401DOI10.21136/AM.2025.0017-25
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