LFU:online

Institut für Mathematik Institut für Mathematik

Univ.-Prof. Mag. Dr. Alexander Ostermann Univ.-Prof. Mag. Dr. Alexander Ostermann, +43 512 507 53810

702701

Einführung in die höhere numerische Mathematik: Numerik von ODEs

VO 2

4

wöch.

jährlich

Deutsch

Absolventinnen und Absolventen dieses Moduls haben einen Überblick über einige aktuelle Fragestellungen der höheren numerischen Mathematik und die Methoden zu deren Behandlung erworben. Weiters haben sie ein vertieftes Verständnis für das Gebiet der numerischen Mathematik erlangt und sind in der Lage, typische Probleme dieses Fachgebietes zu analysieren und zu lösen.

Many time dependent processes in science and engineering are formulated in terms of differential equations. A classical example is given by Newton’s second law which establishes a relation between the (state dependent) forces and the accelerations. Mathematically spoken, this relation is a system of second-order differential equations. Since the majority of such problems cannot be solved analytically, its numerical solution is an important issue. In this lecture, we will introduce Runge-Kutta, extrapolation and multistep methods. We will analyse their error behaviour, derive order conditions, discuss stability and construct particular methods of high order. For stiff problems, we will derive the classical order barriers for multistep methods (due to Dahlquist) and introduce implicit Runge-Kutta. Important concepts such as A-stability, algebraic stability, B-stability and convergence will be discussed in full detail. The lecture will be complemented by exercise sessions.

E. Hairer, S.P. Nørsett, G. Wanner: Solving ordinary differential equations I. Nonstiff problems, 2nd ed., Springer 1992 E. Hairer, G. Wanner: Solving ordinary differential equations II. Stiff and differential-algebraic problems, 2nd ed., Springer 1996

Beginn: 01.10.2012