The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models

Attempts to improve on the basic continuum (hydrodynamic, macroscopic) mode l of traffic flow, as developed in the seminal 1955 paper of Lighthill and Whitham, have largely followed the 1971 work of Payne in retaining the cont inuity equation, but replacing the classical traffic stream model by a "dyn amic traffic stream model" (or "momentum equation"). In the present work it is suggested that, whatever may be the advantages and disadvantages of the Payne models, they should not properly be regarded as the traffic flow ana log of the Navier-Stokes equations of fluid dynamics. Further, the Chapman- Enskog asymptotic expansion in a small parameter is shown to lead to an alt ernate class of models that seem to have a more legitimate claim to that di stinction. Details of this expansion, about the stable-flow equilibria of t he Prigogine-Herman kinetic equation and in the case that the passing proba bility and relaxation time are constant, are presented to orders zero and o ne. The zero-order and first-order expansions correspond respectively to th e Lighthill-Whitham (LWR) model, and to the Lighthill-Whitham model with a diffusive correction. These are suggested to be the correct traffic-flow an alogs of respectively the Euler and Navier-Stokes equations of fluid dynami cs. Results of a numerical simulation for a simple traffic-flow problem sug gest that the diffusive term represents a correction to the LWR model that captures, to some extent, effects stemming from the fact that vehicles actu ally travel at various speeds. (By contrast, Lighthill-Whitham models proce ed as if all vehicles travel at the average speed corresponding to the dens ity of vehicles in their immediate vicinity.) Some further related work is suggested.