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The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik (2014)

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

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank–Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time L2-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed...

Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Nikolaos Bournaveas, Georgios E. Zouraris (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Nikolaos Bournaveas, Georgios E. Zouraris (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

Two Hartree-Fock models for the vacuum polarization

Philippe Gravejat, Christian Hainzl, Mathieu Lewin, Éric Séré (2012)

Journées Équations aux dérivées partielles

We review recent results about the derivation and the analysis of two Hartree-Fock-type models for the polarization of vacuum. We pay particular attention to the variational construction of a self-consistent polarized vacuum, and to the physical agreement between our non-perturbative construction and the perturbative description provided by Quantum Electrodynamics.

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