An analytical shock-fitting algorithm for Lighthill, Whithay & Richards kinematic wave model embedded with linear speed–density relationship

In this paper, we propose a simple and efficient shock-fitting solution algorithm for the LWR model assuming a linear speed–density relationship or parabolic fundamental diagram. The solution is exact if the boundary conditions for density variable on the spatial axis are piecewise linear and those on the time axis are piecewise constant. Discontinuities are explicitly handled. The method utilizes the concept that for a linear speed-density relationship, a linear density variation along the spatial axis remains linear if not interrupted by shocks. Explicit expressions for the nonlinear shock path trajectory between two linear density functions are also derived. Two numerical examples are used to illustrate the effectiveness of the proposed method.