Research Group 3: Scientific Computing

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 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

Office hours:
Mon-Thu 9-12 Uhr

Tel.: +49 721 608 42062

Fax.: +49 721 608 43197

# Geometric Numerical Integration (Summer Semester 2013)

 Lecturer: Prof. Dr. Tobias Jahnke, Lecture (0154100), Problem class (0154200) 2+1

 Lecture: Problem class: Thursday 15:45-17:15 1C-04 Friday 14:00-15:30 Rechnerpool Geb. 01.93 Kronenstr. 32 Raum 101
 Lecturer, Problem classes Prof. Dr. Tobias Jahnke Office hours: Monday, 10 am - 11 am Room 3.042 Kollegiengebäude Mathematik (20.30) Email: tobias.jahnke@kit.edu Dr. Vikram Sunkara Office hours: Room Allianz-Gebäude (05.20) Email: vikram.sunkara@kit.edu Dr. Michael Kreim Office hours: Room Kollegiengebäude Mathematik (20.30) Email: kreim@kit.edu Dr. Tudor Udrescu Office hours: Room Kollegiengebäude Mathematik (20.30) Email: tudor.udrescu@kit.edu

# Aims and scope of the lecture

The numerical simulation of time-dependent processes in science and technology often leads to the problem to solve a system of ordinary differential equations (ODEs) with a suitable method. In many applications it can be shown that the exact solution or flow exhibits certain qualitative or "geometric" properties. For example, it is well-known that the exact flow of a Hamiltonian system is symplectic, and that the energy remains constant along the exact solution although the solution itself changes in time. When the solution or the flow is approximated by a numerical integrator, it is desirable to preserve these geometric properties (at least approximately), because reproducing the correct qualitative behavior is important in most applications. It turns out, however, that many numerical schemes destroy the structure of the solution, and that only selected methods respect the geometric properties of the dynamics. 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 behavior of the method.

We will mainly focus on geometric integrators for Hamiltonian systems of ODEs.

The lecture will be suited for students in mathematics, physics and other sciences with a basic knowledge in ordinary differential equations and Runge-Kutta methods. In particular, students should be familiar with concepts such as, e.g., order, consistency, convergence, A-stability, and so on. The course "Numerische Methoden für Differentialgleichungen" provides a good basis.

# Exercise classes

In the exercise class, students will be asked to write Matlab programs which illustrate the theoretical results presented in the lecture. The exercises can be solved in pairs or alone, and with the assistance of the tutor. Participants are expected to be familiar with Matlab.