Exponential Runge-Kutta methods for stiff kinetic equations

Giacomo Dimarco, Lorenzo Pareschi
(7/10/2010 SIAM J. Num. Anal. 49, 2057-2077, 2011, arXiv:1010.1472v1)

We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The 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 the methods are suitable both for deterministic as well as probabilistic numerical techniques.

Below its the video of a recent lecture I gave on the subject at Newton Institute in Cambridge, UK (you can skip the first minute where the microphone was off).