Abstract
We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.
Highlights
In order to describe traffic flows and to reproduce the formation of congestions, several models based either on Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE) have been proposed
Daganzo [15] pointed out the drawbacks of this approach: the Payne-Whitham model may lead to inconsistent behaviors for the flow, such as vehicles going backwards
For large values of the exponent γ. In this formula Vref > 0 is a reference velocity, to bear in mind the physical meaning of p. This approach is used in fluid mechanics, for modeling certain free boundary problems where bubbles are immersed in a gas [26]
Summary
In order to describe traffic flows and to reproduce the formation of congestions, several models based either on Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE) have been proposed. The velocity offset tends to infinity when ρ → ρ while we get the classical expression p(ρ) ∼ ργ when ρ → 0 Such a pressure law arises in gas dynamics, where it is referred to as the Bethe–. In order to go beyond the simple particulate approach in [7], we wish to develop numerical simulations of the RMAR model (3) with the velocity offset (VO1) for small values of ε.
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