Kinetic decomposition for periodic homogenization problems

We develop an analytical tool which is adept for detecting shapes of oscillatory functions, is useful in decomposing homogenization problems into limit-problems for kinetic equations, and provides an efficient framework for the validation of multi-scale asymptotic expansions. We apply it first to a hyperbolic homogenization problem and transform it to a hyperbolic limit problem for a kinetic equation. We establish conditions determining an effective equation and counterexamples for the case that such conditions fail. Second, when the kinetic decomposition is applied to the problem of enhanced diffusion, it leads to a diffusive limit problem for a kinetic equation that in turn yields the effective equation of enhanced diffusion

Asymptotic Preserving numerical schemes for multiscale parabolic problems

We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale $\varepsilon$. Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behaviour as $\varepsilon \rightarrow 0$, without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, known such homogenization schemes are in general not accurate for both the highly oscillatory regime $\varepsilon \rightarrow 0$ and the non oscillatory regime $\varepsilon \sim 1$. 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.Comment: 7 pages, to appear in C. R. Acad. Sci. Paris; Ser.

Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations

In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schr\"odinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method

Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes

Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing -- i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free-path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially fast as t tends to infinity. This is at variance with the collisionless case discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), pp. 199--221], where the total mass decays as Const./t as the time variable t tends to infinity.Comment: 29 pages, 1 figure, submitted; figure 1 corrected in new versio

Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field

In this paper, we build a Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circonscribed oscillations. The first of these functions has a shape which is close to the shape of the Two-Scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of Two-Scale Asymptotic-Preserving Schemes.Comment: Mathematical Models and Methods in Applied Sciences 00, 00 (2012) 1 --

Multiscale Turbulence Models Based on Convected Fluid Microstructure

The Euler-Poincar\'e approach to complex fluids is used to derive multiscale equations for computationally modelling Euler flows as a basis for modelling turbulence. The model is based on a \emph{kinematic sweeping ansatz} (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zero-mean) motion of fluid parcels at the unresolved scales. Even in the simplest 2-scale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modelling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods.Comment: 1st version, comments welcome! 23 pages, no figures. In honor of Peter Constantin's 60th birthda

Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations

In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two- and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows

Asymptotic preserving schemes for highly oscillatory kinetic equation

This work is devoted to the numerical simulation of a Vlasov-Poisson model describing a charged particle beam under the action of a rapidly oscillating external electric field. We construct an Asymptotic Preserving numerical scheme for this kinetic equation in the highly oscillatory limit. This scheme enables to simulate the problem without using any time step refinement technique. Moreover, since our numerical method is not based on the derivation of the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, and in the highly oscillatory regime as well. Our method is based on a "double-scale" reformulation of the initial equation, with the introduction of an additional periodic variable