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Construction et analyse de conditions aux limites artificielles pour des équations de Schrödinger avec potentiels et non linéarités

Pauline Klein 1, 2
2 CORIDA - Robust control of infinite dimensional systems and applications
IECN - Institut Élie Cartan de Nancy, LMAM - Laboratoire de Mathématiques et Applications de Metz, Inria Nancy - Grand Est
Abstract : The numerical resolution of the Schrödinger equation in exterior domain requires adequate boundary conditions on the boundary of the computational domain. These boundary conditions are directly connected with the potential function of the equation. For the free-potential equation, the exact boundary condition and efficient methods of discretization and implementation are known. The aim of this thesis is to generalize these methods in the case of a potential as general as possible, linear or nonlinear, corresponding to various physical situations. We make the choice to give up exact boundary condition in order to develop a more general method, well-adapted to numerical implementation. Thanks to pseudodifferential calculus, we propose a detailed research of methods taking the potential into account in an artificial boundary condition (ABC). This thesis deals with the case of the one- and two-dimensional Schödinger equation, with a linear or nonlinear potential, and with the case of the one-dimensional stationary equation. The construction of these ABC relies on microlocal analysis and symbolic calculus linked to fractional pseudodifferential operators. The time-discretization is made using discrete convolutions or Padé approximants, and the space discretization relies on linear finite elements. We use the Besse relaxation scheme to solve the nonlinear equation. The mathematical analysis of the constructed boundary conditions shows that in some cases, continuous and semi-discrete a priori estimations are satisfied. Various numerical simulations are implemented in order to test the numerical efficiency of the boundary conditions and to compare them with one another.
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https://tel.archives-ouvertes.fr/tel-01746335
Contributor : Pauline Klein <>
Submitted on : Saturday, January 29, 2011 - 10:32:00 AM
Last modification on : Thursday, March 5, 2020 - 4:52:34 PM
Long-term archiving on: : Tuesday, November 6, 2012 - 12:45:10 PM

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  • HAL Id : tel-01746335, version 2

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Pauline Klein. Construction et analyse de conditions aux limites artificielles pour des équations de Schrödinger avec potentiels et non linéarités. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2010. Français. ⟨NNT : 2010NAN10098⟩. ⟨tel-01746335v2⟩

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