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

The classical Lighthill-Whitham-Richards (LWR) kinematic traffic model is extendedto a unidirectional road on which the maximum density $a(x)$represents roadinhomogeneities, such as variable numbers of lanes, and is allowedto vary discontinuously. The car densityφ = φ(x,t) is then determined by the followinginitial value problem fora scalar conservation law with a spatially discontinuous flux: φₜ + (φ v(φ/a(x))ₓ = 0, φ(x,0)=φ₀(x), x ∈ R, t∈ (0,T), (*)where $v(z)$ is the velocity function. We adapt to (*) a new notionof entropy solutions (Bürger, Karlsen, and Towers[Submitted, 2007]), which involves a Kružkov-typeentropy inequality based on a specific flux connection $(A,B)$, and which we interpret in terms of traffic flow. This conceptis consistent with both the driver's ride impulseand the desire ofdrivers to speed up. We prove that entropy solutionsof type $(A,B)$ are unique. This solution concept also leads to simple, transparent, and unified convergence proofs fornumerical schemes. Indeed, we adjust to (*) new variants of the Engquist-Osher (EO) scheme(Bürger, 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 EO and HW schemes and a related Godunov schemeconverge to the unique entropy solution of type $(A,B)$ of (*). For the Godunov version, this is the first rigorous convergence and well-posedness result, since nounnecessarily restrictive regularity assumptions are imposed on the solution.Numerical experiments for first-order schemes andformally second-orderMUSCL/Runge-Kutta versions are presented.