On the approximation of kinetic equations by moment systems
- The aim of this article 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 arbitrary large speeds of propagation, even for initial data arbitrary close to global eqilibrium.