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32
On TimeSplitting spectral approximations for the Schrödinger equation in the semiclassical regime
 J. Comput. Phys
, 2002
"... In this paper we study timesplitting 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 spatia ..."
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Cited by 84 (47 self)
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In this paper we study timesplitting 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 meshsize constraints are necessary for a uniform L 2approximation of the wave function. The spectral timesplitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L 2approximation 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.
Numerical study of timesplitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes
 SIAM J. Sci. Comput
"... Abstract. In this paper we study the performance of timesplitting spectral approximations for general nonlinear Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant ε is small. The timesplitting spectral approximation under study is explicit, unconditionally stable a ..."
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Cited by 59 (33 self)
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Abstract. In this paper we study the performance of timesplitting spectral approximations for general nonlinear Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant ε is small. The timesplitting spectral approximation under study is explicit, unconditionally stable andconserves the position density in L 1. Moreover it is timetransverse invariant andtimereversible when the corresponding NLS is. Extensive numerical tests are presented for weak/strong focusing/defocusing nonlinearities, for the Gross–Pitaevskii equation, andfor currentrelaxedquantum hydrodynamics. The tests are geared towards the understanding of admissible meshing strategies for obtaining “correct ” physical observables in the semiclassical regimes. Furthermore, comparisons between the solutions of the NLS and its hydrodynamic semiclassical limit are presented.
On the computation of crystalline microstructure
 Acta Numerica
, 1996
"... Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the num ..."
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Cited by 39 (16 self)
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Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.
SelfAveraging in Time Reversal for the Parabolic Wave Equation
 Stochastics and Dynamics
, 2002
"... We analyze the selfaveraging properties of timereversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. ..."
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Cited by 35 (16 self)
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We analyze the selfaveraging properties of timereversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation.
Transport equations for waves in a half space
 Comm. PDE’s
, 1997
"... 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 ectiontransmission relations for plane waves and are well suited for the study of wave propagation in bounded random med ..."
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Cited by 14 (0 self)
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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 ectiontransmission 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
Équilibre instable en régime semiclassique  II: Conditions de BohrSommerfeld
, 1997
"... 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 ..."
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Cited by 12 (2 self)
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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
Global solutions to Maxwell equations in a ferromagnetic medium
"... this paper is to prove the existence of global finite energy solutions to the corresponding Maxwell system (1:2) ..."
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Cited by 11 (0 self)
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this paper is to prove the existence of global finite energy solutions to the corresponding Maxwell system (1:2)
Wigner Functions versus WKBMethods in Multivalued Geometrical Optics
 Anal
, 2003
"... We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the highfrequency asymptotics of such models is reviewed, in particular we highlight the di#culties in crossing caustics when using (timedependent) WKBmethods. Us ..."
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Cited by 9 (2 self)
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We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the highfrequency asymptotics of such models is reviewed, in particular we highlight the di#culties in crossing caustics when using (timedependent) WKBmethods. 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 monokinetic solutions) in the prebreaking regime. Further we show how the Wigner measure approach can be used to analyze highfrequency limits in the postbreaking regime, in comparson with the traditional Fourier integral operator method. Finally we present some illustrating examples.
Extraction of a "Good" Subsequence From a Bounded Sequence of Integrable Functions
, 1995
"... Introduction We consider functions belonging to L 1(\Omega ; ¯; IR d ), where(\Omega ; F ; ¯) is a measured space with ¯ a positive bounded measure. Bounded sequences in L 1 may have weak limit points in the bidual (L 1 ) 00  identified to (L 1 ) 0  out of L 1 . This appears ..."
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Cited by 5 (0 self)
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Introduction We consider functions belonging to L 1(\Omega ; ¯; IR d ), where(\Omega ; F ; ¯) is a measured space with ¯ a positive bounded measure. Bounded sequences in L 1 may have weak limit points in the bidual (L 1 ) 00  identified to (L 1 ) 0  out of L 1 . This appears when uniform integrability (briefly UI) does not hold: see J. Diestel [10, p.45]. The decomposition of such elements in the sum of a function of L 1 and of a non null singular linear form ` s (see on this subject K. Yosida & E. Hewitt [33], [9, Ch.VIII]) has been very efficient in the study of convex integral functionals: at least four papers by R.T. Rockafellar [2124], several Russian works, among t
Weak Convergence and Deterministic Approach to Turbulent Diffusion.
, 1999
"... . 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 ..."
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Cited by 4 (0 self)
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. 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 KortewegdeVries equation and of the Non Li...