Explicit construction of entropy solutions for the Lighthill–Whitham–Richards traffic flow model with a piecewise quadratic flow–density relationship

In this paper we explicitly construct the entropy solutions for the Lighthill–Whitham–Richards (LWR) traffic flow model with a flow–density relationship 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. The proposed model is a generalization of the well-known piecewise linear flow–density relationship in the LWR model. 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 two representative traffic flow cases and also compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme.