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We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale ε . Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behavior as ε → 0 , without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, it is known that such homogenization schemes are in general not accurate for both the highly oscillatory regime ε → 0 and the non-oscillatory regime ε ∼ 1 . In this paper, we introduce an Asymptotic Preserving method based on an exact micro–macro decomposition of the solution, which remains consistent for both regimes. [ABSTRACT FROM AUTHOR]

We construct numerical schemes to solve kinetic equations with anomalous diffusion scaling. When the equilibrium is heavy-tailed or when the collision frequency degenerates for small velocities, an appropriate scaling should be made and the limit model is the so-called anomalous or fractional diffusion model. Our first scheme is based on a suitable micro–macro decomposition of the distribution function, whereas our second scheme relies on a Duhamel formulation of the kinetic equation. Both are Asymptotic Preserving (AP): they are consistent with the kinetic equation for all fixed value of the scaling parameter ε > 0 and degenerate into a consistent scheme solving the asymptotic model when ε tends to 0. The second scheme enjoys the stronger property of being uniformly accurate (UA) with respect to ε . The usual AP schemes known for the classical diffusion limit cannot be directly applied to the context of anomalous diffusion scaling, since they are not able to capture the important effects of large and small velocities. We present numerical tests to highlight the efficiency of our schemes. [ABSTRACT FROM AUTHOR]

Our purpose is to derive a model describing the evolution of particles at various scales following their kinetic energy. Fast particles will be described through a collisional kinetic equation of Boltzmann-BGK type. This equation will be coupled with a fluid model (Euler equations) that describes the evolution of slower particles. The main interest of this approach is to reduce the cost of numerical simulations. To cite this article: N. Crouseilles et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]