Day/time: Mondays, 10:00 am -1:00 pm
Start date: September 10 - December 3, 2001
Location: The Fields Institute, room 210

1. Mathematical Setting [1.1, Chapter 2]

• Solution of perturbed systems
• The defect of a numerical solution
• General error bounds for perturbed systems
• Tight bounds using log-norm

2. General Properties of Numerical Methods [3.1-3.3]

• Classical properties of order/stability/convergence
• Local error, global error, various error bounds

3. Standard Classes of Methods [Chapters 4 and 5]

• One step methods, Taylor series and Runge-Kutta
• Derivation of Runge-Kutta formulas
• Local error estimates for Runge-Kutta formulas
• Multistep methods, Adams formulas
• Derivation of variable step formulas
• Implementation issues for multistep formulas
• Survey of existing software

4. Difficulty of Stiffness [3.4-3.6]

• What is a `stiff problem' and where do they arise
• What are the difficulties/complications that affect computation

5. Special methods for Stiff problems [4.7, 5.1.2, 5.4.3]

• Implicit Runge-Kutta methods
• BDF methods
• Exploiting special problem structure
• Survey of existing software

6. Differential/Algebraic Equations [Chapters 9 and 10]

• Problem structure and classification
• Two basic approaches
• Survey of existing software

7. Delay Differential Equations

• Classification of problems and the associated mathematical properties
• Numerical issues
• Survey of existing software

8. Validated Numerical Methods for ODEs

• Guaranteed error bounds/Interval arithmetic
• Limitations and inherent difficulties

9. Parallel Methods for ODEs

• Special Formulas
• Waveform relaxation
• Other approaches

 

Prerequisites: We assume a solid undergraduate background in mathematics and computer science.
Such a background would normally involve two years of calculus, a year of linear algebra, a year of numerical analysis and exposure to one or more high level programming languages, preferably FORTRAN or C. A mathematical course on the theoretical or analytic properties of differential equations would be helpful, although not essential.

The textbook for the course is: Computer methods for Ordinary Differential Equations and Differential-Algebraic Equations,
U. M. Ascher and L. R. Petzold; SIAM, 1998.