Numerical simulation of macroscopic traffic 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 such as replacing differentials by forward, backward, or central finite differences often are numerically unstable, even in the limit of very small discretizations in space and time. For example, a necessary condition for stability is that the maximum speed at which information about the solution can be propagated by the numerical method must be always greater than the propagation velocity of the exact solution.