Exponential Runge–Kutta Methods for Stiff Kinetic Equations

Abstract

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, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

“…Utilizing on this BGK penalization, Dimarco and Pareschi introduced a class of exponential RungeKutta methods in [10] for the Boltzmann equation, which are exponentially AP in the sense of (10). The starting point is to split the Boltzmann equation (1) into a relaxation step…”

“…A remarkable feature is that, (34) solves f * as a convex combination of positive functions f n , Q(f n ) and M n . Hence the Monte Carlo technique can be applied based on this formulation (see [10]). …”

“…Such schemes are able to remove the stiffness for small ε, and can capture the macroscopic hydrodynamic behavior without numerically resolving the small ε. There have been active recent research activities in developing AP schemes for Boltzmann equation, see [11,12,3,10,28]. We refer to a recent review [23] on AP schemes for kinetic and hyperbolic equations.…”

“…In the work of Dimarco and Pareschi [10] for the Boltzmann equation, a time-splitting approach was introduced, so the convection is solved in a separate step from the collision step. The collision step, using the fact that the local Maxwellian is invariant for the space homogeneous Boltzmann equation, can be solved using the exponential Runge-Kutta method.…”

“…This gives a method in between that of Filbet-Jin and Dimarco-Pareschi, and that combines the advantages of both methods in terms of positivity, asymptotic-preserving and generality. With the positivity this formulation is also suitable for a Monte-Carlo simulation [10]. This paper is organized as follows.…”