G. Dimarco

G. Dimarco, L. Pareschi

We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator, the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally we show some numerical results which confirm the theoretical analysis.

G. Dimarco, L. Pareschi, G. Samaey

We develop a new Monte Carlo method that solves hyperbolic transport equations with stiff terms, characterized by a (small) scaling parameter. In particular, we focus on systems which lead to a reduced problem of parabolic type in the limit when the scaling parameter tends to zero. Classical Monte Carlo methods suffer of severe time step limitations in these situations, due to the fact that the characteristic speeds go to infinity in the diffusion limit. This makes the problem a real challenge, since the scaling parameter may differ by several orders of magnitude in the domain. To circumvent these time step limitations, we construct a new, asymptotic-preserving Monte Carlo method that is stable independently of the scaling parameter and degenerates to a standard probabilistic approach for solving the limiting equation in the diffusion limit. The method uses an implicit time discretization to formulate a modified equation in which the characteristic speeds do not grow indefinitely when the scaling factor tends to zero. The resulting modified equation can readily be discretized by a Monte Carlo scheme, in which the particles combine a finite propagation speed with a time-step dependent diffusion term. We show the performance of the method by comparing it with standard (deterministic) approaches in the literature.