Wir werden uns besonders mit der numerischen Integration von Lagrange und Hamilton Systemen beschäftigen, welche dann zu symplektische Methoden führen. Grundlegende Konzepte wie das Hamilton Prinzip, Symplektizität und Noether’s Theorem werden ebenfalls behandelt. Es wird gezeigt wie diskrete Formulierungen dieser Prinzipien zu einer Klasse von Integratoren führen, den Variationellen Integratoren, die äquivalent zu symplektischen Integratoren sind. Wir werden beweisen warum symplectische Methoden zu genaueren Langzeitintegrationen führen, indem das Konzept der Rückwärtsfehleranalyse vorgestellt wird.

Die Theorie dieser Themen wird von analytischen und praktischen Übungen begleitet, um die Ideen zu verfestigen. Einige dieser Aufgaben umfassen die Implementierung der Integratoren mit Python um Simulationen durchzuführen. Eine Einführung zu Python und notwendigen Bibliotheken ist in den Übungen enthalten.

• E. Hairer, G. Wanner and C. Lubich, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, 2006.
  

• E. Hairer, S. Nørsett, and G. Wanner, Solving ordinary differential equations. I Nonstiff problems. Springer, 1993.
  

• E. Hairer and G. Wanner, Solving ordinary differential equations. II Stiff and differential-algebraic problems. Springer, 2010.
  

• J. E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numerica, 2001.
  

• E. Hairer, C. Lubich and G. Wanner. Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numerica, 2003.
  

• E. Süli and D. F. Mayers, An Introduction to Numerical Analysis. Cambridge University Press, 2003.

Title:

Geometric numerical integration

Credits: 5,0

Prerequisites

Lectures with integrated exercises and programming exercises, 4 SWS, ECTS-Credits: 5.0. Guest students welcomed. Lectures and exercises will be evaluated and credited jointly.

Contents

In this course, numerical methods that preserve the geometric properties of the flow of a differential equation will be presented and analysed.

Basic concepts on numerical integration theory such as consistency and convergence will be introduced. Several numerical integration methods, such as Runge-Kutta, collocation, partitioned methods, composition and splitting methods, will be introduced. The concept of first integrals and conditions for their preservation by some of these methods will be derived and proven.

We will be particularly concerned with the numerical integration of Lagrangian and Hamiltonian systems, which will lead to the introduction of symplectic methods. Basic concepts such as Hamilton's principle, symplecticity, and Noether's theorem will be introduced. It will be shown how a discrete formulation of these principles leads to a class of integrators known as variational integrators which are equivalent to the class of symplectic methods. We will prove why symplectic methods lead to more accurate long-time integration by introducing the concept of backward error analysis.

The theoretical introduction of these topics will be accompanied by analytical and practical exercises to consolidate these ideas. Some of the tasks will involve the implementation of code in Python to run simulations. An introduction to Python and some of the necessary packages will also be contained as part of the sessions.

Literature

• E. Hairer, G. Wanner and C. Lubich, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, 2006.
  

• E. Hairer, S. Nørsett, and G. Wanner, Solving ordinary differential equations. I Nonstiff problems. Springer, 1993.
  

• E. Hairer and G. Wanner, Solving ordinary differential equations. II Stiff and differential-algebraic problems. Springer, 2010.
  

• J. E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numerica, 2001.
  

• E. Hairer, C. Lubich and G. Wanner. Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numerica, 2003.
  

• E. Süli and D. F. Mayers, An Introduction to Numerical Analysis. Cambridge University Press, 2003.