# 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.

# Lecture notes

... are not available any more.

# References

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.