We study the long time motion of fast particles moving through time-dependent random force fields with correlations that decay rapidly in space, but not necessarily in time. The time dependence of the averaged kinetic energy p 2 (t) /2 and mean-squared displacement q 2 (t) is shown to exhibit a large degree of universality; it depends only on whether the force is, or is not, a gradient vector field. When it is, p 2 (t) ∼ t 2/5 independently of the details of the potential and of the space dimension. Motion is then superballistic in one dimension, with q 2 (t) ∼ t 12/5 , and ballistic in higher dimensions, with q 2 (t) ∼ t 2 . These predictions are supported by numerical results in one and two dimensions. For force fields not obtained from a potential field, the power laws are different: p 2 (t) ∼ t 2/3 and q 2 (t) ∼ t 8/3 in all dimensions d ≥ 1.

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test.

Abstract. We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation.