284 research outputs found
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
We present an abstract framework for analyzing the weak error of fully
discrete approximation schemes for linear evolution equations driven by
additive Gaussian noise. First, an abstract representation formula is derived
for sufficiently smooth test functions. The formula is then applied to the wave
equation, where the spatial approximation is done via the standard continuous
finite element method and the time discretization via an I-stable rational
approximation to the exponential function. It is found that the rate of weak
convergence is twice that of strong convergence. Furthermore, in contrast to
the parabolic case, higher order schemes in time, such as the Crank-Nicolson
scheme, are worthwhile to use if the solution is not very regular. Finally we
apply the theory to parabolic equations and detail a weak error estimate for
the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic
heat equation
An effective mass theorem for the bidimensional electron gas in a strong magnetic field
We study the limiting behavior of a singularly perturbed
Schr\"odinger-Poisson system describing a 3-dimensional electron gas strongly
confined in the vicinity of a plane and subject to a strong uniform
magnetic field in the plane of the gas. The coupled effects of the confinement
and of the magnetic field induce fast oscillations in time that need to be
averaged out. We obtain at the limit a system of 2-dimensional Schr\"odinger
equations in the plane , coupled through an effective selfconsistent
electrical potential. In the direction perpendicular to the magnetic field, the
electron mass is modified by the field, as the result of an averaging of the
cyclotron motion. The main tools of the analysis are the adaptation of the
second order long-time averaging theory of ODEs to our PDEs context, and the
use of a Sobolev scale adapted to the confinement operator
The phase shift of line solitons for the KP-II equation
The KP-II equation was derived by [B. B. Kadomtsev and V. I.
Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of
line solitary waves of shallow water. Stability of line solitons has been
proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi,
Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the
local phase shift of modulating line solitons are not uniform in the transverse
direction. In this paper, we obtain the -bound for the local phase
shift of modulating line solitons for polynomially localized perturbations
Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation
The aim of this paper is the accurate numerical study of the KP equation. In
particular we are concerned with the small dispersion limit of this model,
where no comprehensive analytical description exists so far. To this end we
first study a similar highly oscillatory regime for asymptotically small
solutions, which can be described via the Davey-Stewartson system. In a second
step we investigate numerically the small dispersion limit of the KP model in
the case of large amplitudes. Similarities and differences to the much better
studied Korteweg-de Vries situation are discussed as well as the dependence of
the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at
http://www.mis.mpg.de/preprints/index.html
On critical behaviour in systems of Hamiltonian partial differential equations
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture
«La relation de limitation et dâexception dans le français dâaujourdâhui : exceptĂ©, sauf et hormis comme pivots dâune relation algĂ©brique »
Lâanalyse des emplois prĂ©positionnels et des emplois conjonctifs dâ âexceptĂ©â, de âsaufâ et dâ âhormisâ permet dâenvisager les trois prĂ©positions/conjonctions comme le pivot dâun binĂŽme, comme la plaque tournante dâune structure bipolaire. PlacĂ©es au milieu du binĂŽme, ces prĂ©positions sont forcĂ©es par leur sĂ©mantisme originaire dĂ»ment mĂ©taphorisĂ© de jouer le rĂŽle de marqueurs dâinconsĂ©quence systĂ©matique entre lâĂ©lĂ©ment se trouvant Ă leur gauche et celui qui se trouve Ă leur droite. Lâopposition qui surgit entre les deux Ă©lĂ©ments nâest donc pas une incompatibilitĂ© naturelle, intrinsĂšque, mais extrinsĂšque, induite. Dans la plupart des cas (emplois limitatifs), cette opposition prend la forme dâun rapport entre une « classe » et le « membre (soustrait) de la classe », ou bien entre un « tout » et une « partie » ; dans dâautres (emplois exceptifs), cette opposition se manifeste au contraire comme une attaque de front portĂ©e par un « tout » Ă un autre « tout ». De plus, lâinconsĂ©quence induite mise en place par la prĂ©position/conjonction paraĂźt, en principe, tout Ă fait insurmontable. Dans lâassertion « les Ă©cureuils vivent partout, sauf en Australie » (que lâon peut expliciter par « Les Ă©cureuils vivent partout, sauf [quâils ne vivent pas] en Australie »), la prĂ©position semble en effet capable dâimpliquer le prĂ©dicat principal avec signe inverti, et de bĂątir sur une telle implication une sorte de sous Ă©noncĂ© qui, Ă la rigueur, est totalement inconsĂ©quent avec celui qui le prĂ©cĂšde (si « les Ă©cureuils ne vivent pas en Australie », le fait quâils « vivent partout » est faux). NĂ©anmoins, lâanalyse montre quâalors que certaines de ces oppositions peuvent enfin ĂȘtre dĂ©passĂ©es, dâautres ne le peuvent pas. Câest, respectivement, le cas des relations limitatives et des relations exceptives. La relation limitative, impliquant le rapport « tout » - « partie », permet de rĂ©soudre le conflit dans les termes dâune somme algĂ©brique entre deux sous Ă©noncĂ©s pourvus de diffĂ©rent poids informatif et de signe contraire. Les valeurs numĂ©riques des termes de la somme Ă©tant dĂ©sĂ©quilibrĂ©es, le rĂ©sultat est toujours autre que zĂ©ro. La relation exceptive, au contraire, qui nâimplique pas le rapport « tout » - « partie », nâest pas capable de rĂ©soudre le conflit entre deux sous Ă©noncĂ©s pourvus du mĂȘme poids informatif et en mĂȘme temps de signe contraire : les valeurs numĂ©riques des termes de la somme Ă©tant symĂ©triques et Ă©gales, le rĂ©sultat sera toujours Ă©quivalent Ă zĂ©ro
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