A simplified theory of kinematic waves in highway traffic. Part I: General theory

In the theory of ‘kinematic waves,’ as described originally by Lighthill and Whitham in 1955, the evaluation of the shock path is typically rather tedious. Instead of using this theory to evaluate flows or densities, one can use it to evaluate the cumulative flow A(x,t) past any point x by time t. It is shown here how a formal solution for A(x,t) can be evaluated directly from boundary or initial conditions without evaluation at intermediate times and positions. If there are shocks, however, this solution will be multiple-valued. The correct solution, which is the lower envelope of all such formal solutions, will automatically have discontinuities in slope describing the passage of a shock. To evaluate A(x,t) at any particular location x, it is not necessary to follow the actual path of the shock. The solution can be evaluated directly in terms of the boundary data by either graphical or numerical techniques.