Geometric numerical integration (GNI)5 ECTS
(englische Bezeichnung: Geometric numerical integration)
(Prüfungsordnungsmodul: Geometrische numerische Integration)

Lehrveranstaltungen:

• • Geometric numerical integration
    (Vorlesung mit Übung, 4 SWS, Rodrigo Takuro Sato Martin de Almagro et al.)

Inhalt:

• Integration of ordinary differential equations

• Numerical integration

• Conservation of first integrals (linear and quadratic invariants)

• Symplectic integration of Hamiltonian systems

• Variational integrators

• Error analysis

In this lecture, numerical methods that preserve the geometric properties of the flow of a differential equation are presented. First, basic concepts of integration theory such as consistency and convergence are repeated. Several numerical integration methods (Runge-Kutta methods, collocation methods, partitioned methods, composition and splitting methods) are introduced. Conditions for the preservation of first integrals are derived and proven. After a brief introduction into symmetric methods, symplectic integrators for Lagrange and Hamilton systems are considered. Basic concepts such as Hamilton's principle, symplecticity, and Noether's theorem are introduced. A discrete formulation leads to the class of variational integrators which is equivalent to the class of symplectic methods. The symplectictiy leads to a more accurate long-time integration which is proven by concepts of backward error analysis and is demonstrated by means of numerical examples.

Lernziele und Kompetenzen:

Wissen

The students
are familiar with ‘Lagrange systems’ and ‘Hamiltonian systems’ and ‘Hamilton’s principle’
know the terms ‘ordinary differential equation’ and ‘analytic solution’
are familiar with ‘consistency’ and ‘convergence’ of a discrete evolution
know standard integrators to solve ordinary differential equations numerically (Runge-Kutta methods, collocation methods, composition and splitting methods…)
know symmetric integrators
are familiar with the terms ‘first integrals’ and ‘quadratic invariants’
are familiar with Noether’s theorem and symplecticity of the Hamilton flow
know symplectic integrators/variational integrators
know conservation properties of symplectic/variational integrators
are familiar with variational error analysis and backward error analysis

Anwenden

The students
derive Lagrange- and Hamilton’s equations
determine invariants of dynamical systems
implement numerical integrators and solve the ordinary differential equations numerically
analyse the numerical solutions regarding accuracy, conservation of invariants, convergence, symmetry

Literatur:

• 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.

Organisatorisches:

Vertiefungsmodul zum Modul 'Mehrkörperdynamik'

Verwendbarkeit des Moduls / Einpassung in den Musterstudienplan:

1. Berufspädagogik Technik (Master of Education)
   (Po-Vers. 2020w | TechFak | Berufspädagogik Technik (Master of Education) | Gesamtkonto | Wahlpflichtmodule Fachwissenschaft | Wahlpflichtmodule (Vertiefungsmodule) | Geometrische numerische Integration)

Studien-/Prüfungsleistungen:

Geometrische numerische Integration (Prüfungsnummer: 72771)

(englischer Titel: Geometric Numerical Integration)

Prüfungsleistung, mündliche Prüfung, Dauer (in Minuten): 30, benotet, 5 ECTS

Anteil an der Berechnung der Modulnote: 100.0 %

Erstablegung: SS 2022, 1. Wdh.: WS 2022/2023

1. Prüfer:

Sigrid Leyendecker