Dimarco, Giacomo; Pareschi, Lorenzo Exponential Runge-Kutta methods for stiff kinetic equations. (English) Zbl 1298.76150 SIAM J. Numer. Anal. 49, No. 5, 2057-2077 (2011). Summary: 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 nonequilibrium 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. Cited in 3 ReviewsCited in 51 Documents MSC: 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs 35Q20 Boltzmann equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations Keywords:exponential integrators; Runge-Kutta methods; stiff equations; Boltzmann equation; fluid limits; asymptotic preserving schemes Software:Boltzmann PDFBibTeX XMLCite \textit{G. Dimarco} and \textit{L. Pareschi}, SIAM J. Numer. Anal. 49, No. 5, 2057--2077 (2011; Zbl 1298.76150) Full Text: DOI arXiv