Numerical simulation of macroscopic traffic equations

The increasing need for efficient traffic optimization measures is making reliable, fast and robust methods for traffic simulation more and more important. Apart from developing cellular automata models of traffic flow, this need has stimulated studies of suitable numerical algorithms that can solve macroscopic traffic equations based on partial differential equations. The numerical integration of partial differential equations is a particularly difficult task, and there is no generally applicable method. In contrast to ordinary differential equations, the most natural explicit finite difference methods are often numerically unstable, even for very small discretizations of space and time. In general, numerical solutions to partial differential equations require special methods, which work only under certain conditions. Implicit integration methods are usually more stable but they require the frequent solution of linear systems with multidiagonal matrices. In this article, we discuss only explicit methods, because they are useful for the varying boundary conditions found in realistic traffic simulations, where data is continuously fed into the simulation. In addition, explicit methods are more flexible for the simulation of on- and off-ramps or entire road networks.