Many ordinary differential equations have certain structural properties. The exact flow of the differential equation can, for example, be reversible, symplectic, or volume-conserving, and quantities such as the energy, the angular momentum, or the norm of the solution can remain constant although the solution itself changes in time. It is desirable to preserve these geometric properties When the solution or the flow is approximated by a numerical integrator, because reproducing the correct qualitative behaviour is important in most applications. It turns out, however, that many numerical methods destroy the structure of the solution, and only selected methods allow to preserve the geometric properties of the exact flow. 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 behaviour of the method.

We will mainly focus on geometric integrators for Hamiltonian systems.

The lecture will be suited for students in mathematics, physics and other sciences with a basic knowledge in ordinary differential equations.

# 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 StörmerVerlet method.

Acta Numerica 12, 399-450 (2003).