... This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.

In this paper we study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant ε is small. In this regime, the equation propagates oscillations with a wavelength of O(ε), and finite difference approximations require the spatial mesh size h = o(ε) and the time step k = o(ε) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform L 2-approximation of the wave function. The spectral time-splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L 2-approximation of the wave function for k = o(ε) and h = O(ε). Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g., k independent of ε, and h = O(ε)) are admissible for obtaining “correct ” observables. Finally, we address the application to nonlinear Schrödinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies. c ○ 2002 Elsevier Science (USA) 1.

We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation.

We derive boundary conditions for the phase space energy density of acoustic waves in a half space, in the high frequency limit. These boundary conditions generalize the usual re ection-transmission relations for plane waves and are well suited for the study of wave propagation in bounded random media in the radiative transport approximation [15]. The high frequency analysis is based on direct calculations with Fourier integrals in the case of constant coe cients and Wigner measures in general, and it is presented in detail. 1 1

Dans ce travail, nous tudions les valeurs propres de l'oprateur de Schrdinger en dimension 1 qui sont proches d'un maximum local du potentiel. Il fait suite [2] o nous tudiions la concentration des fonctions propres associes. Nous montrons en particulier comment s'e#ectue la transition, dans le cas du double puits symtrique, entre les doublets de valeurs exponentiellement proches et les valeurs rgulirement espaces lorsque l'nergie augmente. 1

this paper is to prove the existence of global finite energy solutions to the corresponding Maxwell system (1:2)

We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the high-frequency asymptotics of such models is reviewed, in particular we highlight the di#culties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the prebreaking regime. Further we show how the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparson with the traditional Fourier integral operator method. Finally we present some illustrating examples.

In this paper we justify by periodic homogenization the double porosity model for immiscible incompressible two-phase flows. The volume fraction of the fissured part and the nonfissured part are kept positive constants and of same order. The scaling is such that, in the final homogenized equations, the less permeable part of the matrix contributes as a nonlinear memory term. For the first part of the proof we use the two-scale convergence as it seems to be appropriate for the problem, even though to work with periodic modulation would be possible. The fact that the equations are nonlinear and degenerated parabolic means that in the final step of the convergence proof on the other hand we have to use periodic modulation instead of two-scale convergence.

For hyperbolic systems in one spatial dimension @ t u+C@ x u = f(u), u(t; x) 2 R d , we study sequences of oscillating solutions by their Young--measure limit ¯ and develop tools to study the evolution of ¯ directly from the Young measure of the initial data. For d 2 we construct a flow mapping S t such that ¯(t) = S t () is the unique Young measure solution for initial value . For d 3 we establish existence and uniqueness of Young measures which have product structure, that is the oscillations in direction of the Riemann invariants are independent. Counterexamples show that neither ¯ nor the marginal measures of the Riemann invariants are uniquely determined from , except if a certain structural interaction condition for f is satisfied. We rely on ideas of transport theory and make usage of the Wasserstein distance on the space of probability measures. 1 Introduction Whenever partial differential equations allow for highly oscillatory solutions it is desirable to find methods to...

. The purpose of this contribution is to show that some of the basic ideas of turbulence can be addressed in a deterministic setting instead of introducing random realizations of the fluid. Weak limits of oscillating sequences of solutions are considered and along the same line the Wigner transform replaces the Kolmogorov definition of the spectra of turbulence. One of the main issue is to show that, at least in some cases, this weak limit is the solution of an equation with an extra diffusion (the name turbulent diffusion appears naturally). In particular for a weak limit of solutions of the incompressible Euler equation (which is time reversible) such process would lead to the appearance of irreversibility. In the absence of proofs, following a program initiated by P. Lax [L], the diffusive property of the limit is analyzed, with the tools of Lax and Levermore [LL] or Jin Levermore and Mc Laughlin [JLM], on the zero dispersion limit of the Korteweg-deVries equation and of the Non Li...