Abstract
We consider the follow-the-leader model for traffic flow. The position of each car $z_i(t)$ satisfies an ordinary differential equation, whose speed depends only on the relative position $z_{i+1}(t)$ of the car ahead. Each car perceives a local density $ρ_i(t)$. We study a discrete traveling wave profile $W(x)$ along which the trajectory $(ρ_i(t), z_i(t))$ traces such that $W(z_i(t)) = ρ_i(t)$ for all $i$ and $t>0$; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at $x\to±∞$ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.
Highlights
Introduction and PreliminariesWe consider a microscopic model for traffic flow
We show that the traveling wave profiles are local attractors for the solution of the FtL model
In this paper we study traveling wave profiles of a particle model for traffic flow, i.e., the follow-the-leader (FtL) ODE models for car positions
Summary
We consider a microscopic model for traffic flow. Let be the length of all the cars, and let zi(t) be the position of ith car at time t. For any given ρ± satisfying (1.15), there exists a profile W (x), unique up to horizontal shifts Such traveling waves are local attractors for the solutions of the FtL model (1.3). Together with Theorem 2.1, we conclude that stationary profiles of W (x), if they exist, are monotonically increasing This corresponds to the upward jumps of the admissible shocks for the conservation law (1.9). We construct approximate solutions to W (x) as a two-point-boundary-value problem, and prove their convergence, establishing the existence of traveling wave profiles. There exists a monotone stationary profile W (x) which satisfies the DDE (1.13) and the “boundary” values lim x→−∞. Once the existence of the profile W (x) is proved, we establish the uniqueness of the solution for the “two-point-boundary-value-problem” for the DDE (1.13). We remark that this is different from a viscous shock profile, where the diffusion is uniform and the profile is odd symmetric about the location of the shock
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