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Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Cancès, Rachida Chakir, Yvon Maday (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this article, we provide a priorierror estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the...

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Cancès, Rachida Chakir, Yvon Maday (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the...

Numerical application of knot invariants and universality of random knotting

Tetsuo Deguchi, Kyoichi Tsurusaki (1998)

Banach Center Publications

We study universal properties of random knotting by making an extensive use of isotopy invariants of knots. We define knotting probability ( P K ( N ) ) by the probability of an N-noded random polygon being topologically equivalent to a given knot K. The question is the following: for a given model of random polygon how the knotting probability changes with respect to the number N of polygonal nodes? Through numerical simulation we see that the knotting probability can be expressed by a simple function of...

Numerical approximation of Knudsen layer for the Euler-Poisson system

Fréderique Charles, Nicolas Vauchelet, Christophe Besse, Thierry Goudon, Ingrid Lacroix–Violet, Jean-Paul Dudon, Laurent Navoret (2011)

ESAIM: Proceedings

In this work, we consider the computation of the boundary conditions for the linearized Euler–Poisson derived from the BGK kinetic model in the small mean free path regime. Boundary layers are generated from the fact that the incoming kinetic flux might be far from the thermodynamical equilibrium. In [2], the authors propose a method to compute numerically the boundary conditions in the hydrodynamic limit relying on an analysis of the boundary layers....

Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena

Nicolas Besse, Norbert J. mauser, Eric Sonnendrücker (2007)

International Journal of Applied Mathematics and Computer Science

We present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus "light waves" are somewhat supressed, which in turn allows thenumerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian...

Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

Numerical study of the systematic error in Monte Carlo schemes for semiconductors

Orazio Muscato, Wolfgang Wagner, Vincenza Di Stefano (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field...

Odd cutsets and the hard-core model on ℤ d

Ron Peled, Wojciech Samotij (2014)

Annales de l'I.H.P. Probabilités et statistiques

We consider the hard-core lattice gas model on ℤ d and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds C d - 1 / 3 ( log d ) 2 , the model exhibits multiple hard-core measures, thus improving the previous bound of C d - 1 / 4 ( log d ) 3 / 4 given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in ℤ d , the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined combinatorial...

On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

In a cholesteric liquid crystal the director field n ( x , y , z ) tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz...

On a mathematical model for the crystallization of polymers

Maria Pia Gualdani (2003)

Bollettino dell'Unione Matematica Italiana

We consider a mathematical model proposed in [1] for the cristallization of polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. The model includes a constraint W e ⁢ q on the crystal volume fraction. Essentially, the model is a system of both second order and first order evolutionary partial differential equations with nonlinear terms which are Lipschitz continuous, as in [1], or Hölder continuous, as in [3]. The main novelty here is the...

On a model of rotating superfluids

Sylvia Serfaty (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an energy-functional describing rotating superfluids at a rotating velocity ω , and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical ω above which energy-minimizers have vortices, evaluations of the minimal energy as a function of ω , and the derivation of a limiting free-boundary problem.

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