### Geometric Numerical Integration (Summer Semester 2020)

- Lecturer: Prof. Dr. Tobias Jahnke
- Classes: Lecture (0154100), Problem class (0154200)
- Weekly hours: 3+1

In spite of the restrictions caused by the COVID-19 pandemia, the lecture and the tutorial will take place. As long as "classical" classes are not possible, however, both the lecture and the tutorial will be replaced by a digital format:

- I will publish a revised version of my lecture notes for download. Additional comments and explanations will be given in lecture-like videos which can be downloaded from a central KIT address. The address will be communicated in due time.
- The exercises for the tutorial will be posted in ILIAS. Solutions of these exercises will be published after the deadline. You have the possibility to submit your answers to the exercises in order to get feedback. (Please use the interface provided in ILIAS to hand in your solutions.)
- Additional information will be provided in ILIAS. In particular, there will be a forum where you can ask questions concerning the lecture or the videos.
- My office hours are suspended until further notice, but feel free to contact me by e-mail and, if necessary, by Skype.

This unusual way of learning and teaching is a considerable challenge for students, tutors and lecturers. Benny Stein and I will do our best to handle the situation as good as we can. Please let me know if you have problems or suggestions for improvements.

Schedule | |||
---|---|---|---|

Lecture: | Tuesday 15:45-17:15 | SR 3.061 | Begin: 21.4.2020, End: 21.7.2020 |

Friday 8:00-9:30 (every 2nd week) | SR 2.066 | ||

Problem class: | Friday 8:00-9:30 (every 2nd week) | SR 2.066 | Begin: 24.4.2020, End: 24.7.2020 |

Lecturers | ||
---|---|---|

Lecturer | Prof. Dr. Tobias Jahnke | |

Office hours: Monday, 10 am - 11 am | ||

Room 3.042 Kollegiengebäude Mathematik (20.30) | ||

Email: tobias.jahnke@kit.edu | ||

Problem classes | MSc Benny Stein | |

Office hours: Monday, 10-12 a.m. (lecture period) and by appointment | ||

Room 3.045 Kollegiengebäude Mathematik (20.30) | ||

Email: benny.stein@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 with a suitable method. In many applications it can be shown that the exact flow exhibits certain qualitative or "geometric" properties. For example, it is well-known that the flow of a Hamiltonian system is symplectic, and that the energy remains constant along the 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, because reproducing the correct qualitative behavior is important in many applications. It turns out, however, 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.

# Tutorial

In the tutorial students will solve exercises in order to get familiar with the concepts and results presented in the lecture. In some of these exercises, students will be asked to write programs in Matlab or Python in order to test the behaviour of numerical methods.

# Prerequisites

The lecture will be suited for Master 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. Moreover, participants are expected to be familiar with Matlab or Python. In some of the exercises discussed in the tutorial, students will be asked to write short programs in order to test the behaviour of numerical methods.

Both the lecture and the exercise classes will be given in English.

# ILIAS

All materials are provided in ILIAS. There you will find a discussion forum and the lecture recordings, too.