# Geometric Numerical Integration I W17/18

*They throw geometry out the door, and it comes back through the window.*

-- H. G. Forder, Auckland 1973

*Applied mathematics is not engineering.*

-- Paul Halmos, Mathematics Tomorrow 1981

Dates | Description | Prerequisites | Lecture Notes | Exercises | References

## Dates

Lecture: Tuesday 14:15-15:45, 02.10.011 (Rechnerraum)

Exercises: Thursday 16:15-17:45, 03.13.010 (18.01.2018 in 00.08.053)

First Lecture: 17.10.2017

Last Lecture: 06.02.2018

No Lecture on: 31.10., 26.12., 02.01.

Exercises: 25.10., 08.11., 29.11., 11.01., 18.01., 25.01., 01.02., 08.02.

## Description

In this course, we will cover basic techniques of structure-preserving or geometric numerical integration for ordinary differential equations. We will review the theoretical background of the continuous theory, that is some geometry background, basics of Lagrangian and Hamiltonian dynamics, variational principles, the concept of symplecticity, and how the Noether theorem connects symmetries and conservation laws. We will discuss basic symplectic integrators for Hamiltonian systems and explore strategies for proving symplecticity. We will construct a discrete counterpart to Lagrangian mechanics, including Hamilton's action principle and the Noether theorem, leading to variational integrators. We will cover energy- and integral-preserving methods. We apply basic technqiues from backward error analysis in order to gain a better understanding of the behaviour of some structure-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 stateoftheart 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.

In the end, there will be an oral examination of approximately 30 minutes. 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. No helping material is allowed.

## Prerequisites

Prerequisites are

- Linear Algebra (MA1101),
- Ordinary Differential Equations (MA2005),
- Numerics of Ordinary Differential Equations (MA2304).

Advantegeous are

- Modeling and Simulation of ODEs (MA9803),
- Numerics of Dynamical Systems (MA3333),
- Basic Differential Geometry (MA2204),
- Differential Forms (MA5016),
- Theoretical Mechanics (PH0005).

Familiarity with basic numerical methods (approximation, interpolation, numerical quadrature) is highly recommended. Knowledge of basic differential geometry (manifolds, vector fields, differential forms, tangent and cotangent bundles) and classical mechanics (Lagrangian and Hamiltonian dynamics, action principles, symplecticity, Noether theorem) is of advantage but not required. All important concepts will be reviewed at the beginning of the course.

## Lecture Notes

Script (modified 07.11.2017)

1. Introduction (use Acrobat Reader for animations)

2. Some Numerical Analysis Background

3. Some Geometry Background

4. Lagrangian and Hamiltonian Dynamics

5. Symplectic Integrators

6. Volume Preserving Integrators

7. Backward Error Analysis

8. Variational Integrators

9. Integral Preserving Integrators

## Exercises

1. Review Part 1 (25.10.2017)

2. Review Part 2 (08.11.2017)

3. Symplectic Integrators (Solutions)

4. Volume Preserving Integrators

5. Variational Integrators

6. Integral Preserving Integrators

## References

- Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration. Springer, 2006. (eBook)
- Benedict Leimkuhler and Sebastian Reich. Simulating Hamiltonian Dynamics. Cambridge University Press, 2005. (eBook)
- Jerrold E. Marsden and Matthew West. Discrete Mechanics and Variational Integrators. Acta Numerica Volume 10, 357-514, 2001. (Journal)