A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model

Authors:
Mengping Zhang;Chi-Wang Shu;George C. K. Wong;S. C. Wong
Affiliations:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China;Division of Applied Mathematics, Brown University, Providence, RI;Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China;Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China
Venue:
Journal of Computational Physics
Year:
2003

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Abstract

In this paper, we present a high-order weighted essentially non-oscillatory (WENO) scheme for solving a multiclass extension of the Lighthill-Whitham-Richards (LWR) model. We first review the multi-class LWR model and present some of its analytical properties. We then present the WENO schemes, which were originally designed for computational fluid dynamics problems and for solving hyperbolic conservation laws in general, and demonstrate how to apply these to the present model. We found through numerical experiments that the WENO method is vastly more efficient than the low-order Lax-Friedrichs scheme, yet both methods converge to the same solution of the physical model. It is especially interesting to observe the small staircases in the solution which are completely missed out, because of the numerical viscosity, if a lower-order method is used without a sufficiently refined mesh. To demonstrate the applicability of this new, efficient numerical tool, we study the multi-class model under different parameter regimes and traffic stream models. We consider also the convergence of the multi-class LWR model when the number of classes goes to infinity. We show that the solution converges to a smooth profile without staircases when the number of classes increases.