Explicit Construction of Entropy Solutions for the Lighthill-Whitham-Richards Traffic Flow Model with a Non-Smooth Flow-Density Relationship

Abstract

In this paper we explicitly construct the entropy solutions for the Lighthill-Whitham-Richards (LWR) traffic flow model with a flow-density relationship q(ρ) which is piecewise quadratic, continuous, concave, but not differentiable at the junction points where two quadratic polynomials meet, and with piecewise linear initial condition and piecewise constant boundary conditions. As observed traffic flow data can be well fitted with such continuous piecewise quadratic functions, the explicitly constructed solutions provide a fast and accurate solution tool which may be used for predicting traffic or as a diagnosing tool to test the performance of numerical schemes. We implement these explicit entropy solutions for three representative traffic flow cases and also compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme.

Real-Time Solution of Velocity-Based First-Order Continuum Traffic Model with Finite Element Method

“…More recently, researchers have shifted their attention to special cases in which the flow-density relationship is not smooth. Lu et al considered the case in which the flow-density relationship is either nondifferentiable or discontinuous at some points and provide solution algorithms for these cases that are based on the entropy principle (10,11).…”