WI305106 - Numerieke Wiskunde 2/Numerical Analysis 2

Numerical Modelling of Dynamical Systems

Goal: To introduce the theory and application of numerical methods for initial value problems in ordinary and stochastic differential equations.

Meets:
Weeks 36 - 49, except week 43
(Sep. 2 - Dec. 4, 2008 except 21, 23 Oct.)
Tuesday 15:00-17:00, REC-P.015/A
Thursday 9:00-11.00, REC-P.015/B

Summary:
This course provides a thorough introduction to the numerical treatment of continuous dynamical systems, both deterministic and stochastic. We make the acquaintence of three classes of methods: Runge-Kutta, multistep, and vector-field splitting methods; and analyze their approximation properties and convergence. For integrations on intervals, we discuss important aspects of asymptototic behavior, such as linear stability of equilibria and preservation of invariant sets. As a unique aspect, we give special attention to the parallels between our subject and the theories of continuous and discrete dynamical systems. Such methods enjoy wide-spread use in applications such as mathematical physics, biology and finance, from which we draw examples.

Follow this link for an illustrative example of the different dynamical properties of numerical methods.

Topics:
The course will cover the following topics:

• Continuous and discrete dynamical systems
• Numerical approximation of ordinary and stochastic differential equations
• Runge-Kutta, multistep, and splitting methods
• Equilibria and concepts of stability
• Integration on manifolds

Lecturer:
Jason Frank
Centrum Wiskunde & Informatica (CWI)
Kruislaan 413, Room M259
tel: +31 20 5924096
e-mail:
www: http://www.cwi.nl/~jason

Final Exam (4 Dec. 2008)

Lecture Notes
The lecture notes appear here and are updated upon occasion.

• Chapter 1. Modelling (updated 5 Nov 2008)
• Chapter 2. Simulation (updated 5 Nov 2008)
• Chapter 3. Solutions (updated 5 Nov 2008)
• Chapter 4. The Matrix Exponential (updated 5 Nov 2008)
• Chapter 5. Flow Maps and Dynamical Systems (updated 5 Nov 2008)
• Chapter 6. Convergence of One-Step Methods (updated 5 Nov 2008)
• Chapter 7. Collocation Methods (updated 5 Nov 2008)
• Chapter 8. Runge-Kutta Methods (updated 5 Nov 2008)
• Chapter 9. Equilibrium Points (updated 5 Nov 2008)
• Chapter 10. Stability (updated 8 Dec 2008)
• Chapter 11. Interlude: Linear Multistep Methods (updated 5 Nov 2008)
• Chapter 12. Reversibility and Composition Methods (updated 5 Nov 2008)
• Chapter 13. Splitting Methods (updated 5 Nov 2008)
• Chapter 14. First Integrals (updated 5 Nov 2008)
• Chapter 15. Constrained Dynamics (updated 5 Nov 2008)
• Chapter 16. Symplectic Flows and Maps and Volume Preserving Methods (updated 18 Nov 2008)
• Chapter 17. Backward Error Analysis (updated 25 Nov 2008)
• Appendix. Implementational Issues (updated 26 Sep 2008)

Sample Matlab programs: (Directory listing)