Explicit Construction Of Entropy Solutions For The LWR Traffic Flow Model

In this thesis we explicitly construct the entropy solution for the Lighthill-Whitham-Richards (LWR) traffic flow model with a flow-density relationship q(p) which is piecewise quadratic, concave and with piecewise linear initial condition and piecewise constant boundary conditions. We first discuss the situation that q(ρ) is continuous, but not differentiable at the junction points where the quadratic polynomials meet. We then discuss the situation for dis-continuous q(p). The existence and uniqueness of entropy solutions for such conservation laws with discontinuous fluxes are not known mathematically. We have used the approach of explicitly constructing the entropy solutions to a sequence of approximate problems in which the flow-density relationship q(ρ) is continuous but tends to the discontinuous flux when a small para-meter in this sequence tends to zero. The limit of the entropy solutions for this sequence is explicitly constructed and is considered to be the entropy solution associated with the discontinuous flux. At last we implement these explicit entropy solutions for several representative traffic flow cases, compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme, and discuss the results from traffic flow perspectives.