Fallstudien der Mathematischen Modellbildung, part 3: Asymptotic methods for perturbation problems

The goal of this lecture is to introduce some fundamental notions and techniques used in the asymptotic analysis of perturbation problems. Such problems are called singular if they undergo a change in their mathematical structure as the perturbation parameter ε tends to zero. A solution of the reduced problem (ε = 0) coincides with the limit solution of the full problem as ε --> 0 only if the perturbation is regular. It is the subject of asymptotic analysis to find approximate solutions of the full problem that are valid uniformly for 0 < ε <= ε₀, even if the perturbation is singular. Singular perturbation problems usually arise at the most critical (and interesting) regimes of physical modeling - their analysis and ultimate resolution has often lead to major advances in a specific field of science. In the first part of this course we focus on some basic principles and examples in the context of ordinary differential equations: we introduce the principle of dominant balance and discuss boundary layers, the WKB method, the method of (variational) averaging and the method of multiple scales. The guiding-center approximation of plasma physics is considered as a generic example of nonlinear perturbation theory. In the second part we extend our analysis to partial differential equations and present Prandtl's boundary layer for the Navier-Stokes equation. Moreover, we elaborate on macroscopic limits of kinetic equations in the strongly collisional regime, leading to fluid models of reduced dimensionality.

Outline of the lectures

• Regular and singular perturbations: basic notions
• The process of "non-dimensionalization"
• Asymptotic expansions: order functions, order of a function, asymptotic series, ...
• Regular perturbations of nonlinear initial-value problems: Gronwall lemma, guiding-center problem
• Singular perturbations of linear ODEs: boundary layers, averaging, technique of multiple scales, WKB, ...
• Extensions to PDEs: Prandtl's boundary layer (Navier-Stokes), anisotropic transport and diffusion
• Macroscopic limits of kinetic equations: drift-diffusion, hydrodynamic and adiabatic limit, collision operators, H-theorem

Exercise classes

The lectures will be complemented with an exercise class where the notions developed in the lecture will be applied to concrete examples.

News

• First lecture: Monday, 7. 1. 2019, 16:15 - 17:45 at MI Hörsaal 3 (00.06.011)
• First exercise class: Wednesday, 9. 1. 2019, 17:00 - 18:30 at CH 21010 Hans-Fischer-Hörsaal (5401.01.101K)

Literature

• C. Bender and S. Orszag, Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory, Springer Science & Business Media (2013).
• J. Cousteix and J. Mauss, Asymptotic analysis and boundary layers, Springer Science & Business Media (2007).
• P. Degond, L. Pareschi and G. Russo, Modeling and Computational Methods for Kinetic Equations, Springer Science+Business Media, LLC (2004).
• W. Eckhaus, Asymptotic analysis of singular perturbations, Elsevier (2011).
• A. Nayfeh, Perturbation methods, John Wiley & Sons (2008).
• J. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, Springer (2007).
• E. de Jager and J.F. Furu, The theory of singular perturbations, Elsevier (1996).

Lecturers

• Lectures and exercises: Dr. Stefan Possanner.

Lecture notes

• Lecture notes Feb. 5

Exercise sheets

• Exercise sheet 1: First perturbation problems and order functions. --> Solutions
• Exercise sheet 2: Asymptotic expansions (AEs). --> Solutions
• Exercise sheet 3: Regular perturbations of nonlinear IVPs.
• Exercise sheet 4: Application of the K.B.M. Theorem to weakly nonlinear oscillations.

Examination