### Splitting methods for evolution equations (Summer Semester 2020)

- Lecturer: Prof. Dr. Tobias Jahnke
- Classes: Lecture (0160800), Problem class (0160810)
- 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 | |||
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Lecture: | Monday 11:30-13:00 | SR 3.061 | Begin: 20.4.2020, End: 20.7.2020 |

Friday 11:30-13:00 (every 2nd week) | SR 3.061 | ||

Problem class: | Friday 11:30-13:00 (every 2nd week) | SR 3.061 | Begin: 24.4.2020, End: 24.4.2020 |

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 | Dr. 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 |

# Scope

Splitting methods are a very popular class of integrators solving time-dependent ordinary or partial differential equations numerically. The underlying idea is to decompose the differential equation into two or more subproblems which can be solved exactly or more efficiently, and to construct an approximation of the full problem by a suitable composition of the flows of the subproblems.

After a short introduction to splitting methods for ordinary differential equations, the lecture will focus on splitting methods for partial differential equations such as, e.g., linear and nonlinear Schrödinger-type equations and parabolic problems. Special attention will be given to the convergence analysis, in particular to the relation between the order of convergence and the regularity of the data. This will require some results from semigroup theory, which will be provided in the lecture. If you can attend Roland Schnaubelt's lecture Evolution Equations or if you have already had a similar course, this would be optimal but not a necessary condition to attend my lecture.

# Prerequisites

Students are expected to be familiar with ordinary differential equations, Runge-Kutta methods (construction, order, convergence analysis, stability) and Sobolev spaces (definition, basic properties, Sobolev embeddings).

# ILIAS

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