Nonstandard second-order formulation of the LWR model

Wen-Long Jin

doi:10.1080/21680566.2019.1617803

Abstract

ABSTRACTIn this study, we present an equivalent second-order formulation of the LWR model based on Phillips’ model, in which the acceleration rate equals a relaxation term with an infinitesimal relaxation time. We convert the model into a continuum car-following model and demonstrate its equivalence to that of the LWR model. We demonstrate that the nonstandard second-order model is stable, but the original Phillips model is not. We derive conditions for the nonstandard model to be forward-traveling and collision-free, prove that the collision-free condition is consistent with but more general than the CFL condition, and demonstrate that only anisotropic and symplectic Euler discretization methods lead to physically meaningful solutions. Numerically, the nonstandard second-order model has the same shock and rarefaction wave solutions as the LWR model for both Greenshields and triangular fundamental diagrams; for a non-concave fundamental diagram, the collision-free condition, but not the CFL condition, yields physically meaningful results.

Highlights

The seminal LWR model describes the evolution of traffic density, k(t, x), speed, v(t, x), and flow-rate, q(t, x) in a time-space domain (Lighthill and Whitham, 1955; Richards, 1956) and can be derived from the following rules (Hereafter we omit (t, x)): R1
It was shown in (LeVeque, 2001) that, in the LWR model for night traffic, the characteristic wave speed can be larger than vehicles’ speeds, but anisotropic solutions still exist if one converts the LWR model into a car-following model, except that such anisotropic solutions are different from those obtained with the traditional Oleinik entropy condition for hyperbolic conservation laws
We present a nonstandard second-order model by replacing the relaxation time in Phillips’ model by a hyperreal infinitesimal number

Summary

Introduction

The seminal LWR model describes the evolution of traffic density, k(t, x), speed, v(t, x), and flow-rate, q(t, x) in a time-space domain (Lighthill and Whitham, 1955; Richards, 1956) and can be derived from the following rules (Hereafter we omit (t, x)): R1. In many second-order continuum models, the characteristic wave speeds can be higher than vehicles’ speeds; this has led to the conclusion that such models are not anisotropic, as information can travel faster than vehicles (Daganzo, 1995a) It was shown in (LeVeque, 2001) that, in the LWR model for night traffic, the characteristic wave speed can be larger than vehicles’ speeds, but anisotropic solutions still exist if one converts the LWR model into a car-following model, except that such anisotropic solutions are different from those obtained with the traditional Oleinik entropy condition for hyperbolic conservation laws. With the nonstandard method, we will be able to derive conditions to guarantee forward-traveling and collision-free solutions in the car-following model for a second-order continuum model.

A nonstandard second-order model

Nonstandard Phillips’ model

Instability of Phillips’ model

Equivalence and stability

Equivalence between the nonstandard second-order model and the LWR model

Stability

Forward-traveling and collision-free properties

Conditions for forward-traveling and collision-free

Other discretization methods

Arithmetic central difference method

Numerical examples

Greenshields fundamental diagram