Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model

Abstract

The classical Lighthill-Whitham-Richards (LWR) kinematic traffic modelis extended to a unidirectional road on which the maximum density a(x)represents road inhomogeneities, such as variable numbers of lanes, andis allowed to vary discontinuously. The car density phi = phi(x, t) isthen determined by the following initial value problem for a scalarconservation law with a spatially discontinuous flux:phi t + (phi v(phi/a(x))(x) = 0, phi(x, 0) = phi(0)(x), x is an elementof R, t is an element of (0, T), (x)where v(z) is the velocity function. We adapt to ({*}) a new notion ofentropy solutions (Burger, Karlsen, and Towers {[}Submitted, 2007]),which involves a Kruzkov-type entropy inequality based on a specificflux connection (A, B), and which we interpret in terms of traffic flow.This concept is consistent with both the driver's ride impulse and thedesire of drivers to speed up.We prove that entropy solutions of type (A, B) are unique. This solutionconcept also leads to simple, transparent, and unified convergenceproofs for numerical schemes. Indeed, we adjust to ({*}) new variants ofthe Engquist-Osher (EO) scheme (Burger, Karlsen, and Towers{[}Submitted, 2007]), and of the Hilliges-Weidlich (HW) scheme analyzedby the authors {[}J. Engrg. Math., to appear]. It is proven that the EOand HW schemes and a related Godunov scheme converge to the uniqueentropy solution of type (A, B) of ({*}). For the Godunov version, thisis the first rigorous convergence and well-posedness result, since nounnecessarily restrictive regularity assumptions are imposed on thesolution. Numerical experiments for first-order schemes and formallysecond-order MUSCL/Runge-Kutta versions are presented.