Riv. Mat. Univ. Parma, Vol. 10, No. 2, 2019

Claudia Negulescu [a]

Analytical and numerical aspects of the linear and non-linear Schrödinger equation

Pages: 351-446
Received: 2 October 2018
Accepted in revised form: 3 April 2019
Mathematics Subject Classification (2010): 34L40, 34E05, 34E20, 35Q41, 42B20, 65L20, 65M06, 81S22.
Keywords: Schrödinger equation, Multi-scale problems, Asymptotic analysis, Asymptotic-preserving schemes.
Authors address:
[a]: Université Paul Sabatier, Institut de Mathématique de Toulouse, 118 route de Narbonne, 31062 Toulouse, France

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Abstract: The subject-matter of these lecture notes, presented in 2016 at the Ravello Summer School on ''Mathematical Physics'' as well as in 2018 at the Università degli Studi di Napoli Federico II, as a PhD Course at the Doctoral School ''Mathematics and Applications'', is a presentation of several numerical techniques for the simulation of the linear respectively non-linear Schrödinger equation, arising in a variety of physical and biological contexts. In particular, the author is interested in the design and ensuing mathematical and numerical study of multi-scale schemes, as for ex. Asymptotic-Preserving schemes, for the resolution of the Schrödinger equation in several regimes, as for example the semi-classical regime. The lectures are based on articles, which were chosen to illustrate different techniques in the construction of such efficient numerical schemes for singular perturbation problems. However, the here developped techniques can also be applied for various other singular perturbation problems and these notes can serve as an introduction to more elaborate works on the subject

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