Geometric Numerical Integration of ODEs W15/16

They throw geometry out the door, and it comes back through the window.
-- H. G. Forder, Auckland 1973

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

Exams

In case you plan to take the oral exam, please drop me a mail to michael.kraus@ipp.mpg.de.

Dates

Monday and Wednesday 10:15, 02.08.020, Seminarraum M11

First Lecture: 12.10.2015
Last Lecture: 03.02.2016
No Lecture on: 26.10., 28.10., 23.11., 25.11., 23.12.
Exercises: 14.10., 21.10., 11.11., 16.12., 20.01., 03.02.

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-preserving methods and discuss how to deal with constrained systems and external forcing. We apply basic technqiues from backward error analysis in order to gain a better understanding of the behaviour of some structure-preserving methods.

In the first exercises, we give a short introduction into Python, Cython and the IPython notebook, which provides a framework for efficient implementation, prototyping and testing of the discussed algorithms. We provide practical tips on implementation issues, for instance of nonlinear implicit methods, including how to obtain good starting approximations and how to reduce rounding errors in order to obtain fast convergence and good long-time accuracy.

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.

In the end, there will be an oral examination of approximately 30 minutes. Students demonstrate knowledge of important geometric structures of ordinary differential equations. They can select suitable numerical methods which preserve these structures. They explain the underlying ideas of the various geometric integrators, sketch their construction and apply methods for their analysis. No helping material is allowed.

Prerequisites

Prerequisites are

Advantegeous are

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

Download script (modified 25.02.2016). Because of image copyright this material may not be redistributed.

  1. Introduction (modified 02.11.2015)
  2. Some Geometry Background (modified 25.02.2016)
  3. Lagrangian and Hamiltonian Dynamics (modified 25.02.2016)
  4. Symplectic Integrators (modified 16.12.2015)
  5. Backward Error Analysis (modified 15.12.2015)
  6. Variational Integrators (modified 26.01.2016)
  7. Volume-Preserving Methods (modified 27.01.2016)
  8. Integral Preserving Methods (modified 27.01.2016)
  9. Symplectic Energy-Momentum-Preserving Methods (modified 31.01.2016)

Exercises

  1. Introduction to Python (Notebooks)

  2. Implementation of Nonlinear Implicit Methods

    • Fixed-point and Newton Iteration
    • Starting Approximations
    • Reducing Rounding Errors
  3. Symplectic Integrators (Exercise Sheet 1, Solutions, Notebook, HTML)

  4. Backward Error Analysis and Variational Integrators (Exercise Sheet 2, Solutions)

  5. Volume Preserving and Integral Preserving Methods (Exercise Sheet 3, Solutions, Notebook, HTML)

References

Main References
Differential Geometry
Geometric Mechanics
Numerical Analysis
Scientific Computing with Python
High Performance Computing with Python
The Python Programming Language
Research Articles and Reviews