Abstract
We consider a semi-discrete car-following model and the macroscopic Aw-Rascle model for traffic flow given in Lagrangian form. The solution of the car-following model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous system, we allow vacuum in the initial data. By using properties of the semi-discrete model, we show that this solution of the hyperbolic system is stable in the $L^1$-norm.
Highlights
We consider a semi-discrete model for traffic flow proposed by Aw, Klar, Materne and Rascle in [1], τkδ (t) =1 δ vkδ+1(t) − vkδ (t) where wkδ = vkδ + Q(τkδ). (1)wkδ (t) = R τkδ(t), wkδ (t)The functions τkδ(t) and vkδ(t) are the inverse density and velocity of car k, respectively
After investigating the above model, we show that the solution converges to a weak entropy solution of the macroscopic Aw–Rascle model for traffic flow [2] as δ → 0
In this paper we show that the weak entropy solution of the hyperbolic system of conservation laws obtained as a limit of the car-following model, is stable in the L1-norm
Summary
We consider a semi-discrete model for traffic flow proposed by Aw, Klar, Materne and Rascle in [1], τkδ (t). We derive the Aw–Rascle model by considering the semi-discrete car-following model, and show that the limit is a weak entropy solution of this system. The limit (τ, w) is a weak entropy solution of the macroscopic Aw–Rascle model (2) with Cauchy initial data (τ, w). Assume wδ converges to w in L1loc(R) and let τδ be given by xδ = ∂τδ/∂y where xδ(y) is a function which converges to x(y) = ∂τδ/∂y pointwise a.e. the solution τ δ(y, t) of the semi-discrete car-following model (1) converges to a weak entropy solution of the macroscopic Aw–Rascle model (2) with Cauchy initial data (τ, w), and the limit (τ, w) is in ΓV. This discontinuity in the conversion between Eulerian and Lagrangian coordinates explains the disparity
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