EXPONENTIAL RUNGE-KUTTA METHODS FOR STIFF KINETIC EQUATIONS

• Published in: SIAM journal on numerical analysis Vol. 49; no. 5/6; pp. 2057 - 2077
• Main Authors: DIMARCO, GIACOMO, PARESCHI, LORENZO
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Boltzmann equation; Coefficients; Convexity; Decomposition; Entropy; Equilibrium; Exact sciences and technology; Kinetic equations; Kinetics; Mathematical analysis; Mathematics; Measure and integration; Molecules; Monte Carlo methods; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Runge Kutta method; Sciences and techniques of general use; Boltzmann equation; exponential integrators; Differential equation; fluid limits; Decomposition method; 35Q20; Non linear equation; Runge-Kutta methods; Integrodifferential equation; Numerical stability; 65M75; Entropy solution; Transcendental equation; Transport equation; 65L06; Integration; Collision operator; 65L04; Asymptotic behavior; Runge Kutta method; Non linear system; Equation system; Extrapolation; Numerical analysis; Sufficient condition; Multistep method; stiff equations; Kinetic equation; Algebraic equation; asymptotic preserving schemes
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/100811052

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.