79 research outputs found
An improved transient algorithm for the simulation of a resonant tunneling diode
40 pagesA fast algorithm is proposed for the simulation of the transient evolution of a resonant tunneling diode by a multiscale approach. The problem is modeled by the time-dependent Schrödinger-Poisson system. By decomposing the wave function into a non resonant part and a resonant part, the fast algorithm is designed by combining standard finite difference method for the Schrödinger-Poisson equation with proper time-dependent and/or nonlinear transmission boundary condition. In addition, with a suitable interpolation of the non resonant one, an accurate and fast algorithm is presented for the computation of the resonant part via the projection method. With this two scale decomposition, the new numerical method can save the computational time significantly
High density limit of the stationary one dimensional Schrödinger-Poisson system
International audienceThe stationary one dimensional Schrödinger-Poisson system on a bounded interval is considered in the limit of a small Debye length (or small temperature). Electrons are supposed to be in a mixed state with the Boltzmann statistics. Using various reformulations of the system as convex minimization problems, we show that only the first energy level is asymptotically occupied. The electrostatic potential is shown to converge towards a boundary layer potential with a profile computed by means of a half space Schrödinger-Poisson system
Diffusion and guiding center approximation for particle transport in strong magnetic fields
International audienceThe diffusion limit of the linear Boltzmann equation with a strong magnetic field is performed. The giration period of particles around the magnetic field is assumed to be much smaller than the collision relaxation time which is supposed to be much smaller than the macroscopic time. The limiting equation is shown to be a diffusion equation in the parallel direction while in the orthogonal direction, the guiding center motion is obtained. The diffusion constant in the parallel direction is obtained through the study of a new collision operator obtained by averaging the original one. Moreover, a correction to the guiding center motion is derived
Central limit theorems for additive functionals of ergodic Markov diffusions processes
We revisit functional central limit theorems for additive functionals of
ergodic Markov diffusion processes. Translated in the language of partial
differential equations of evolution, they appear as diffusion limits in the
asymptotic analysis of Fokker-Planck type equations. We focus on the square
integrable framework, and we provide tractable conditions on the infinitesimal
generator, including degenerate or anomalously slow diffusions. We take
advantage on recent developments in the study of the trend to the equilibrium
of ergodic diffusions. We discuss examples and formulate open problems
A corrector theory for diffusion-homogenization limits of linear transport equations
This paper concerns the diffusion-homogenization of transport equations when
both the adimensionalized scale of the heterogeneities and the
adimensionalized mean-free path \eps converge to 0. When \alpha=\eps, it is
well known that the heterogeneous transport solution converges to a homogenized
diffusion solution. We are interested here in the situation where
0<\eps\ll\alpha\ll1 and in the respective rates of convergences to the
homogenized limit and to the diffusive limit. Our main result is an
approximation to the transport solution with an error term that is negligible
compared to the maximum of and \frac\eps\alpha. After establishing
the diffusion-homogenization limit to the transport solution, we show that the
corrector is dominated by an error to homogenization when \alpha^2\ll\eps and
by an an error to diffusion when \eps\ll\alpha^2.
Our regime of interest involves singular perturbations in the small parameter
\eta=\frac\eps\alpha. Disconnected local equilibria at need to be
reconnected to provide a global equilibrium on the cell of periodicity when
. This reconnection between local and global equilibria is shown to
hold when sufficient {\em no-drift} conditions are satisfied. The Hilbert
expansion methodology followed in this paper builds on corrector theories for
the result developed in \cite{NBAPuVo}.Comment: 25 page
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