Tentative lecture topics (and hours for each topic) for
CSC2302H for the Winter term, 2014.
Note that section numbers in square brackets refer to relevant
sections of the text for the course.


0. Overview of Problem Area and Course Organization [1.1, 3.1.1] (2 hours)
   ================================================


1. Mathematical Background and Numerical Implications [1.1, Chapter 2] (4 hours)
   ========================================================

     -Solution of perturbed ODEs
     -The defect of a numerical (or approximate) solution
     -General error bounds for perturbed systems
     -Tight bounds using log-norm

2. General Properties of Numerical Methods [3.1-3.3] (4 hours)
   =======================================

     -Classical properties of order/stability/convergence
     -Local error, global error and typical error bounds

3. Standard Classes of Methods [Chapters 4 and 5] (8 hours)
   ===========================

     -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] (2 hours)
   =======================

     -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] (2 hours)
   ==================================

     -Implicit Runge-Kutta methods
     -BDF methods
     -Exploiting special problem structure 
     -Survey of existing software for stiff problems

6. Differential/Algebraic Equations [Chapters 9 and 10] (1 hour)
   ================================

     -Problem structure and classification
     -Two basic approaches
     -Survey of existing software for DAE problems

7. Delay Differential Equations (2 hours) 
   ============================

     -Classification of problems and the associated mathematical properties
     -Numerical issues
     -Survey of existing software for DDE problems

8. Other Related Software Tools  for Investgating IVPs (1 hour)
   ============================

      -global error estimates
      -condition number estimates
      -parameter estimates
      -sensitivity estimates (of the solution and of the parameters)

9. Numerical Methods on High Performane Systems (1 hour)
   ============================================

     - multicore and parallel architectures
     - Special Formulas
     - Waveform relaxation
     - Other approaches