MTH
452-
552
: NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS - Winter 2013
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RECALL that textbook exercies are here
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(1-2) 1/7-9: Introduction, review of ODEs and well-posedness. Read 5.1,5.2
HW 1 due Monday, 1/14.
(3-4) 1/11-14: Difference approximations to derivatives. Read Chapter 1.
(5) 1/16: Basic schemes for solving ODEs: Euler, Heun,
Trapezoidal. Read 5.3-
HW 2 due Wednesday, 1/23.
(5) 1/18: LTE (local truncation error analysis). H
(6) 1/23: Lorentz system example. (Extra credit
opportunity). Compare BE and FE solutions. How to implement implicit
methods for nonlinear problems: pitfalls of simple iteration vs
Newton's method.
(7-8) 1/25-28: [Read the rest of Chapter 5]. Butcher tableaus. Choice
of time step based on a-posterior error analysis.
HW 3 due Friday, 2/1.
(9-10) 1/30-2/1: Worksheet on proving consistency of multi-step and
multi-stage methods. (Extra credit opportunity).
(11-12) 2/4-6: Global error and error propagation. Zero-stability (Chapter 6) and convergence.
(13) 2/8: Review for exam.
(14) Midterm Monday, 2/11.
(15-17) 2/13-15, 2/18. Absolute stability (Chapter 7).
HW 4 due Monday, 2/25.
(18-19) 2/20-22. Deriving growth factors for one-step methods,
plotting stability regions. Predictor-corrector P(EC)^nuE methods.
BDF* methods. Order stars.
(20-21) 2/25-27. Chapter 8: A-stability and L-stability. Nonlinear problems
and stability. Linear and nonlinear systems and the choice of time
step. How to linearize a problem.
HW 5 due Wednesday, 3/6.
(22) 3/1. Wrap-up initial value problems. How to select a method
and the time-step. IMEX methods and example of a convection-diffusion
problem (viscous Burgers).
(23-24) 3/4-6. Start BVP (Chapter 2). Well- and ill-posedness with
various source terms and boundary conditions. Discretization
and consistency. Grid norms.
HW 6 due Wednesday, 3/13.
(25) 3/8. Example of consistency.
(26) 3/11. Convergence plots and tables. Stability proof in 2-norm.
(27-28) 3/13-15. Review. Worksheet on choosing time steps.
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