Investigating the Dynamics of Traffic Flow on a Highway in a Class of Discontinuous Functions

Abstract

This work is devoted to finding a solution of Riemann problem for the first order nonlinear partial equation which describes the traffic flow on highway. When $\rho _{\ell }>\rho _r$, the solution is presented as a piecewise continuous function, where $\rho _{\ell }$ and $\rho _r$ are the densities of cars on the left and right side of the intersection respectively. On the contrary case, a shock of which the location is unknown beforehand arises in the solution. In this case, a special auxiliary problem is introduced, the solution of which makes it possible to write the exact solution showing the locations of shock. For the realization of the proposed method, the parameters of the flow are also found.