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Vorlesung
Analytical and Numerical Aspects of Multiscale Problems Arising in Quantum Mechanics or Plasma Physics
Zusammenfassung: The central theme of this course is the introduction of multiscale numerical schemes for the resolution of singularly perturbed problems arising in the description of sys tems composed of N charged particles evolving in an electrostatic or electromagnetic field. The main aim is to present, via some selected examples, how to treat numer ically problems containing disparate scales in time and/or space, scales which have to be captured in an adequate manner by the scheme without so much numerical ef fort, meaning if possible with low computational times and low memory requirements. Multiscale phenomena emerge very often around us and the schemes presented here can be applied in various (other) physical contexts, as for example in the study of the dynamics of neutral gases or other multiparticle interacting systems, however the focus of these lectures is set on applications coming from quantum mechanics and plasma physics. The techniques I shall present exploit somehow the disparity of the scales in order to develop performant schemes sharing the efficiency of macroscopic models and the accuracy of microscopic ones. In the process of development of such multiscale methods, the detailed understanding of the relations between the micro scopic and the macroscopic worlds is fundamental (multiscale modelling, scaling pro cedure) and analytical techniques such as asymptotic analysis and limits are essential.
Inhalte: The introduction is thought to introduce the physical context of the problems I shall treat in these lectures, as well as to explain the main ideas of multiscale modelling, analysis and numerics, such that to put things in perspective. Chapter 1 introduces the basic features of the Schrödinger equation and some notions of quantum mechanics. Chapter 2 deals with the stationary, linear Schödinger equation in the semiclassical regime; the concerned application is the simulation of the electron transport in a RTD; the numerical scheme is a multiscale FEM, based on the WKBapproximation. Chapter 3 presents briefly different existing numerical schemes for the timedependent, linear Schrödinger equation in the semiclassical regime. Chapter 4 deals with the cubic nonlinear Schödinger equation, applied to the modeling of BECs; the numerical scheme is based on splitting procedures. Chapter 5 introduces the FokkerPlanck equation, arising often in thermonuclear fusion plasma descriptions, and underlines some of its properties. Chapter 6 is concerned finally with a performant numerical scheme for the VlasovPoissonFokkerPlanck equation in the low electrontoion mass ratio regime; the numerical scheme is a spectral method, based on the construction of a wellsuited orthonormal Hermite basis set.
Sprache: The written course material as well as the lecture will be in English.
Literatur:

C. Le Bris "Systemes multiéchelles. Modélisation et simulation", Springer Verlag BerlinHeidelberg, 2005.

T. Cazenave, "Semilinear Schrödinger equation", Courant Lecture Notes in Mathematics AMS 10, 2003.

F. F. Chen, "Plasma Physics and controlled fusion", Springer Verlag NewYork ,2006.

W. E "Principles of multiscale modeling", Cambridge university press, 2011.

Ch. Lubich, "From quantum to classical molecular dynamics: reduced models and numerical analysis", European Mathematical Society, 2008.

A. Quarteroni, "Numerical Models for Differential Problems", Springer Verlag BerlinHeidelberg, 2017.

E. Sonnendrücker, "Computational Plasma Physics", [MA4304]
ECTS: 3 CP
Erforderliche Vorkentnisse: Familiarity with ODEs, PDEs, and numerical methods.