Geometric Numerical Integration of ODEs W15/16
-- H. G. Forder, Auckland 1973
Dates | Description | Prerequisites | Lecture Notes | Exercises | References
In case you plan to take the oral exam, please drop me a mail to email@example.com.
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.
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.
- Linear Algebra (MA1101),
- Ordinary Differential Equations (MA2005),
- Numerics of Ordinary Differential Equations (MA2304).
- 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.
Download script (modified 25.02.2016). Because of image copyright this material may not be redistributed.
- Introduction (modified 02.11.2015)
- Some Geometry Background (modified 25.02.2016)
- Lagrangian and Hamiltonian Dynamics (modified 25.02.2016)
- Symplectic Integrators (modified 16.12.2015)
- Backward Error Analysis (modified 15.12.2015)
- Variational Integrators (modified 26.01.2016)
- Volume-Preserving Methods (modified 27.01.2016)
- Integral Preserving Methods (modified 27.01.2016)
- Symplectic Energy-Momentum-Preserving Methods (modified 31.01.2016)
Introduction to Python (Notebooks)
Implementation of Nonlinear Implicit Methods
- Fixed-point and Newton Iteration
- Starting Approximations
- Reducing Rounding Errors
Symplectic Integrators (Exercise Sheet 1, Solutions, Notebook, HTML)
Backward Error Analysis and Variational Integrators (Exercise Sheet 2, Solutions)
Volume Preserving and Integral Preserving Methods (Exercise Sheet 3, Solutions, Notebook, HTML)
- 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)
- John M. Lee. Introduction to Smooth Manifolds. Springer, 2013. (eBook)
- Loring W. Tu. An Introduction to Manifolds. Springer, 2011. (eBook)
- Gerardo F. Torres del Castillo. Differentiable Manifolds: A Theoretical Physics Approach. Birkhäuser, 2012. (eBook)
- Mikio Nakahara. Geometry, Topology and Physics. CRC Press, 2003.
- Theodore Frankel. The Geometry of Physics. Cambridge University Press, 2011.
- Michael Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish, 1999.
- Ralph Abraham, Jerrold E. Marsden and Tudor S. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer, 1988. (eBook)
- Jorge V. José and Eugene J. Saletan. Classical Dynamics. Cambridge University Press, 1998.
- Vladimir I. Arnol’d. Mathematical Methods of Classical Mechanics. Springer, 1989. (eBook)
- Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and Symmetry. Springer, 1999. (eBook)
- Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. Addison-Wesley, 1987. (eBook)
- Peter Deuflhard, Andreas Hohmann. Numerical Analysis in Modern Scientific Computing. Springer, 2003. (eBook)
- Peter Deuflhard, Folkmar Bornemann. Scientific Computing with Ordinary Differential Equations. Springer, 2002. (eBook)
- Walter Gautschi: Numerical Analysis: An Introduction. Birkhäuser, 2012. (eBook)
- Ernst Hairer, Syvert P. Nørsett and Gerhard Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 1993. (eBook)
- Ernst Hairer and Gerhard Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 1996. (eBook)
- Alfio Quarteroni, Riccardo Sacco and Fausto Saleri. Numerical Mathematics. Springer, 2007. (eBook)
- Kendall Atkinson and Weimin Han. Theoretical Numerical Analysis: A Functional Analysis Framework. Springer, 2009. (eBook)
Scientific Computing with Python
- Robert Johansson. Lectures on Scientific Computing with Python. (GitHub)
- Anthony Scopatz and Kathryn D. Huff. Effective Computation in Physics. O'Reilly, 2015. (Safari Books)
- Hans Petter Langtangen. A Primer on Scientific Programming with Python. Springer, 2014. (eBook)
High Performance Computing with Python
- Kurt W. Smith. Cython. O'Reilly, 2015. (Safari Books)
- Micha Gorelick, Ian Ozsvald. High Performance Python. O'Reilly, 2014. (Safari Books)
- Fernando Doglio, Mastering Python High Performance. Packt Publishing, 2015. (Safari Books)
The Python Programming Language
- Mark Lutz. Learning Python. O'Reilly, 2013. (Safari Books)
- Luciano Ramalho. Fluent Python. O'Reilly, 2015. (Safari Books)
- Paul Gries, Jennifer Campbell, Jason Montojo. Practical Programming: An Introduction to Computer Science Using Python 3. Pragmatic Bookshelf, 2013. (Safari Books)
Research Articles and Reviews
- Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration Illustrated by the Störmer–Verlet Method. Acta Numerica 12, 399-450, 2003. (Journal)
- Robert I. McLachlan and G. Reinout W. Quispel. Splitting Methods. Acta Numerica 11, 341-434, 2002. (Journal)
- Sebastian Reich. Backward Error Analysis for Numerical Integrators. SIAM Journal on Numerical Analysis 36, 1549-1570, 1999. (Journal)
- Ernst Hairer. Backward Error Analysis for Multistep Methods. Numerische Mathematik 84, 199-232, 1999. (Journal)