Geometric Numerical Integration 1

In this course, we will cover basic techniques of structure­preserving or geometric numerical integration for ordinary differential equations.
1. Lagrangian and Hamiltonian Dynamics
2. Symplectic Integrators
3. Backward Error Analysis
4. Volume­-Preserving Methods
5. Variational Integrators
6. Energy­-Preserving Methods

After successful completion of the module, students are able to recognize various geometric structures present in many ordinary differential equations. They have an overview of state-of-­the-art numerical integration methods which preserve these structures and are able to select and implement suitable methods depending on the equations at hand and the desired conservation properties. Participating students are able to proof the conservation properties of the presented methods, either by direct computation, the discrete Noether theorem, or backward error analysis.

Linear Algebra and Discrete Structures 1 (MA1101)
Ordinary Differential Equations (MA2005)
Numerics of Ordinary Differential Equations (MA2304)

Lectures + Exercises
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

chalk board

-­ Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration. Springer, 2006.
- Benedict Leimkuhler, Sebastian Reich. Simulating Hamiltonian Dynamics. Cambridge University Press, 2005.
- Jerrold E. Marsden and Matthew West. Discrete Mechanics and Variational Integrators. Acta Numerica 10, 2001.

Requirement for the successful completion of the module is to pass an oral exam. In the exam, students show their knowledge of important geometric structures of ordinary differential equations and appropriate numerical methods, which preserve these structures. They show the ability to analyse and prove the conservation properties of geometric numerical integrators as well as the ability to apply such integrators to problems from scientific computing.