On the time discretization of kinetic equations in stiff regimes
Seminar Room 1, Newton Institute
We review some results concerning the time discretization of kinetic equations in stiff regimes and their stability properties. Such properties are particularly important in applications involving several lenght scales like in the numerical treatment of fluid-kinetic regions. We emphasize limitations presented by several standard schemes and focus our attention on a class of exponential Runge-Kutta integration methods. Such methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation operators of BGK type. For Boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.