On the approximation of the Fokker-Planck equation by moment systems

The aim of this paper is to show that moment approximations of kinetic equations based on a maximum-entropy approach can suffer from severe drawbacks if the kinetic velocity space is unbounded. As example, we study the Fokker-Planck equation where explicit expressions for the moments of solutions to Riemann problems can be derived. The quality of the closure relation obtained from the maximum-entropy approach as well as the Hermite/Grad approach is studied in the case of five moments. It turns out that the maximum-entropy closure is even singular in equilibrium states while the Hermite/Grad closure behaves reasonably. In particular, the admissible moments may lead to arbitrarily large speeds of propagation, even for initial data arbitrary close to global eqilibrium.