Applications of Mathematics, first online, pp. 1-26
Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle
Lukáš Vacek, Chi-Wang Shu, Václav Kučera
Received January 17, 2025. Published online June 12, 2025.
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.
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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
