Abstract
We revisit the entropy formulation and the wave-front tracking construction of physically admissible solutions of the Aw-Rascle and Zhang (ARZ) ``second-order'' model for vehicular traffic. A Kruzhkov-like family of entropies is introduced to select the admissible shocks. This tool allows to define rigorously the appropriate notion of admissible weak solution and to approximate the solutions of the ARZ model with point constraint. Stability of solutions w.r.t. strong convergence is justified. We propose a finite volumes numerical scheme for the constrained ARZ, and we show that it can correctly locate contact discontinuities and take the constraint into account.
Highlights
Any macroscopic vehicular traffic model expresses the conservation of the number of vehicles along a highway without entrances or exists with the PDE ρt + (ρ v)x = 0, where ρ is the density and v is the velocity of the vehicles
A particular attention was paid to incorporating point constraints of the kind (ρ v)|x=xi ≤ qi (t ) into the Lighthill-Whitham and Richards (LWR) model
As suggested in [13], the unilateral constraint is taken into account via an additional singular “compensation term” in the Kruzhkov entropy inequalities supported at x = xi
Summary
Any macroscopic vehicular traffic model expresses the conservation of the number of vehicles along a highway without entrances or exists with the PDE ρt + (ρ v)x = 0, where ρ is the density and v is the velocity of the vehicles. In addition to this PDE, the Lighthill-Whitham and Richards (LWR) model, [26, 28], assumes that v = V (ρ). We refer to [5, 13, 15, 18, 29, 30] for the introduction, the fundamental analytic results and a simple approximation strategy for the locally constrained LWR model.
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