Arbeitsgruppe 3: Wissenschaftliches Rechnen

Sekretariat
Kollegiengebäude Mathematik (20.30)
Zimmer 3.039

Zimmer 3.039
Englerstr. 2
Kollegiengebäude Mathematik (20.30)
76131 Karlsruhe

Karlsruher Institut für Technologie (KIT)
Fakultät für Mathematik
Institut für Angewandte und Numerische Mathematik
Arbeitsgruppe 3: Wissenschaftliches Rechnen
Englerstr. 2
Kollegiengebäude Mathematik (20.30)
D-76131 Karlsruhe

Öffnungszeiten:
Mo-Do 9-12 Uhr

Tel.: 0721 608 42062

Fax.: 0721 608 43197

# Geometric numerical integration - SoSe 2009 (Sommersemester 2009)

 Dozent: Prof. Dr. Tobias Jahnke Vorlesung (1608) 2 Mathematics and other sciences (ab 3. Semester)

 Vorlesung: Freitag 11:30-13:00 Seminarraum 34
 Dozent Prof. Dr. Tobias Jahnke Sprechstunde: Montag, 10:00 - 11:00 Uhr Zimmer 3.042 Kollegiengebäude Mathematik (20.30) Email: tobias.jahnke@kit.edu

Many ordinary differential equations have certain structural properties. The exact flow of the differential equation can, for example, be reversible, symplectic, or volume-conserving, and quantities such as the energy, the angular momentum, or the norm of the solution can remain constant although the solution itself changes in time. It is desirable to preserve these geometric properties When the solution or the flow is approximated by a numerical integrator, because reproducing the correct qualitative behaviour is important in most applications. It turns out, however, that many numerical methods destroy the structure of the solution, and only selected methods allow to preserve the geometric properties of the exact flow. These methods are called geometric numerical integrators.

In this lecture we will investigate

• why certain methods are (or are not) geometric numerical integrators,
• how to construct geometric numerical integrators,
• which properties are conserved, and in which sense,
• how structure conservation is related to the long-time error behaviour of the method.

We will mainly focus on geometric integrators for Hamiltonian systems.

The lecture will be suited for students in mathematics, physics and other sciences with a basic knowledge in ordinary differential equations.

# Literaturhinweise

Ernst Hairer, Christian Lubich, and Gerhard Wanner:
Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Second edition, Springer, 2006.

Ernst Hairer, Christian Lubich, and Gerhard Wanner:
Geometric numerical integration illustrated by the StörmerVerlet method.
Acta Numerica 12, 399-450 (2003).