Traveling waves for nonlocal models of traffic flow

Johanna Ridder, Wen Shen

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

We consider several nonlocal models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with nonlocal flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. (See definitions 1.1 and 1.2, respectively.) We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition.

Original languageEnglish (US)
Pages (from-to)4001-4040
Number of pages40
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume39
Issue number7
DOIs
StatePublished - Jul 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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