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Exponential Runge-Kutta methods for stiff kinetic equations. (English) Zbl 1298.76150

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.

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

Software:

Boltzmann
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