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

Applications of Mathematics (2025)

  • Issue: 3, page 311-339
  • ISSN: 0862-7940

Abstract

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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.

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 I: $L^2$ stability." Applications of Mathematics (2025): 311-339. <http://eudml.org/doc/299987>.

@article{Vacek2025,
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.},
author = {Vacek, Lukáš, Shu, Chi-Wang, Kučera, Václav},
journal = {Applications of Mathematics},
keywords = {traffic flow; discontinuous Galerkin method; Godunov numerical flux; $L^2$ stability},
language = {eng},
number = {3},
pages = {311-339},
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 I: $L^2$ stability},
url = {http://eudml.org/doc/299987},
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 I: $L^2$ stability
JO - Applications of Mathematics
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 3
SP - 311
EP - 339
AB - 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.
LA - eng
KW - traffic flow; discontinuous Galerkin method; Godunov numerical flux; $L^2$ stability
UR - http://eudml.org/doc/299987
ER -

References

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