# Geometric Numerical Integration (Summer Semester 2016)

Lecturer: | Prof. Dr. Tobias Jahnke |
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Classes: | Lecture (0154100), Problem class (0154200) |

Weekly hours: | 3+1 |

#### NEWS

24.8.2016, 9:00 - 18:00 | Oral exams: Geometric numerical integration (1) |

Lecture: | Thursday 8:00-9:30 | SR 3.61 |
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Wednesday 8:00-9:30 (every 2nd week) | SR 3.61 | |

Problem class: | Wednesday 8:00-9:30 (every 2nd week) | -1.031 |

Lecturer | Prof. Dr. Tobias Jahnke |
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Office hours: Monday, 10 am - 11 am | |

Room 3.042 Kollegiengebäude Mathematik (20.30) | |

Email: tobias.jahnke@kit.edu | |

Problem classes | Dipl.-Math. oec. Marcel Mikl |

Office hours: | |

Room 3.038 Kollegiengebäude Mathematik (20.30) | |

Email: marcel.mikl@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 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 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.

A short description of the course can be downloaded here.

# Evaluation

The results of the evaluation can be found here.

# Numerical simulations

- GNI illustrated by the pendulum
- Simulation of the outer solar system (without movie)

# 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, at home or in class, and with the assistance of the tutor. Participants are expected to be familiar with MATLAB. As KIT has a Campus License for MATLAB, all students can download and install the software. More information here.

The exercise class will take place on Wednesday at 8:00 in the computer pool -1.031. This time slot is shared with the lecture, i.e. the exercise class and lecture will alternate on Wednesdays. The exercise class will probably take place on

27 April, 11 May, 25 May, 8 June, 22 June, 6 July, 20 July

Problem Sheet 01 (The Mathematical Pendulum)

Problem Sheet 02 (Symplectic Maps) Set of Points 1 Set of Points 2

Problem Sheet 03 (Solar System) Data Table

Problem Sheet 04 (Multiple Pendulum)

# References

Jesus Maria Sanz-Serna and Mari Paz Calvo: *Numerical Hamiltonian problems.*

Number 7 in Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1994.

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).

Benedict Leimkuhler and Sebastian Reich: *Simulating Hamiltonian dynamics.*

Cambridge monographs on applied and computational mathematics 14, Cambridge University Press, 2004.