A Multiclass Car-Following Rule Based on the LWR Model

Abstract

The Lighthill-Whitham and Richards WR)(L model ([14], [17]) is well knownfor its simplicit,y parsimony and its robustness to replicate basic trac fea-tures. However, it considers trac as a homogeneous ow. This can be aserious limitation when the trac stream is composed of radically dierentvehicle classes, such as cars and heavy trucks near an uphill grade.Extensive research has been conducted to introduce heterogeneity in theWRLmodel; e.g. [18], [19], [20], [3], [1], [15], [2]. All these extensions are based onthe same principle: disaggregating the heterogeneous trac ow into homo-geneous and continuum vehicle classes that obey a conservation law with aspecic fundamental diagram (FD). These models are solved numerically inEulerian coordinates with methods such as the Godunov scheme [9]. However,these schemes are known to be very diusive for hyperbolic systems of con-servation laws [2].Recent developments in trac ow theory have led to ecient numericalschemes for solving the WRL model. They are derived in Lagrangian frame-work rather than in traditional Eulerian one; see for example [16], [4], [12],[13]. Additionally,ariationalv theory [6], [7], [8] and its extension in Lagrangiancoordinates [13] make it possible to prove that these schemes are exact for theWRL model when the FD is triangular. This is an important leap forwardsince current methods introduce numerical errors that can be devastating inpractice. The aim of this paper is to extend the framework in [13] in order toincorporate multiple vehicle types, each one with a dierent car-following rule.In this w,ay the free-ow speed, the jam density and the wave-speed can bedened for each individual vehicle class. Note that the one-class car-followingrule has already been coupled with a lane-changing model [10] and thus theproposed extension is fully compatible with the latter.The sketch of the paper is as follows: section 1 will recall the Lagrangianformulation of the WRL model and its numerical resolution using (i) the Go-dunov scheme and (ii) the ariativonal theory. Section 2 will introduce the