MAT 5187 TOPICS IN APPLIED MATHEMATICS
(Numerical Methods for Ordinary Differential Equations)
**UNDER CONSTRUCTION**
Session du printemps / Spring Term: 2007.05.02 - 2007.07.18 (4h/weeks - 3 cr.)

**COURSE DESCRIPTION:**
Méthodes numériques pour équadif. / Numerical methods for ode's.

**PREQUISITES:** Third year mathematics.

**INSTRUCTOR:** Rémi VAILLANCOURT

room: 301G KED (585 King Edward)
tel.: 562-5800 ext 3533
e-mail: remi@uottawa.ca
home page: http://www.site.uottawa.ca/~remi
**Office hours:** Preferably by email or by appointment.
**TIME AND PLACE :**

Mondays and Wednesdays, 10:00-12:00, VNR 462

**MAIN TEXT:**
*Numerical methods in ordinary differential equations*, J.D. Lambert, Wiley, Chichester, 1991.
Interested students can order their own text from Amazon or other book sellers.

**OTHER TEXTS:**

*Computational methods in ordinary differential equations*, J.D. Lambert, Wiley, London, 1973.
*Numerical methods for ordinary differential equations*, J.C. Butcher, Wiley, Chichester, 2003.
*The numerical analysis of ordinary differential equations*, J.C. Butcher, Wiley, Chichester, 1987.
*Solving ordinary differential equations I, Nonstiff problems*, E. Hairer, S.P. Norsett, G. Wanner, 2nd rev. ed., Springer, Berlin, 2000.
*Solving ordinary differential equations II, Stiff and differentiala-algebraic problems*, E. Hairer, G. Wanner, Springer, Berlin, 1991.
*Computing solution of ordinary differential equations , The initial value problem*, L.F. Shampine, M.K. Gordon, W.H. Freeman, San Francisco, 1975.
*Solving ordinary differential equations II, Stiff and differentiala-algebraic problems*, E. Hairer, G. Wanner, Springer, Berlin, 1991.
*Behind and Beyond the MATLAB ODE Suite*, R. Ashino, M. Nagase, R. Vaillancourt, Computers Math. Appl., vol. 40 (2000) 491-512. Download pdf:
Behind.pdf
Research papers on HB, HBO and HBT methods.
**LECTURE NOTES:**

Download pdf notes on numerical methods for ode's in French based on Lambert's 1973 book :
MNEquadif.pdf.
**MARKING SCHEME. TO BE DISCUSSED:**

Assignments (AS): 20%
Takehome final (THF): 40%
Final exam (FE): 40%
**COURSE OBJECTIVES:**

The course will try to serve all students.
In particular, it will lay down the preliminaries for students who are preparing their
theses on new numerical methods for ode's.

**COURSE OUTLINE - NOT FINALIZED YET:**

Mathematical preliminaries
Linear multistep methods
Predictor-corrector methods
Runge--Kutta methods
Hermite--Birkhoff (HB), HB-Obrechkoff, and HB-Taylor methods
Continuous interpolants
Stiff differential equations
**APPROXIMATE ORGANIZATION OF LECTURES - NOT FINALIZED YET:**

June 6 at 10:00 Yu ZHANG will present his pre-defence video. Everyone is invited
to ask questions during the presentation. Everyone is invited to the defence on
Friday, June 15th at 1:30 p.m. in STE 1010. Since the topics is new Hermite-Birkhoff-Obrechkoff
generalized multistep methods, you are invited to preview the video and the paper
in the Canadian Applied Mathematics Quarterly, by downloading these pdf files.
YuSlide.pdf
CAMQ.pdf
June 11-15: STUDY BREAK
FINAL EXAM Take-home part. Due Wednesday, 25 July 2007.
Question 1: Show that convergence implies consistency of numerical methods for totally stable ode's.
Question 2: Lambert's 1991, p. 51, exercise 3.2.5*.
Suppose that the order of y_{n+r} is sufficiently high.
Question 3: Prove that Matlab's ode23 is made of a method of order 3 to advance the step
and a method of order 2 to monitor the local truncation error.
Question 4: Prove that k-step BDF methods of order k are unstable for k
greater than 6.
Download pdf:
Takehomeexam.pdf
BDFinstability.pdf
FINAL EXAM Wednesday, 18, July 2007, 09:00-12:00, MRT 221 (Open book, all calculators allowed). Typical questions.
Question 1: Use backward differences to derive ONE of the three methods of order 3:
3-step Adams-Bashforth, or 2-step Adams-Moulton, or 3-step BDF.
Question 2: Find the domain of absolute stability of the following methods:
1-step AB (Euler), 1-step AM (trapezoidal), and 1-step BDF.
Question 3: Show that for all four-stage explicit Runge-Kutta methods of order
4, c_4=1.
Question 4: Show that for all semi-implicit Runga-Kutta methods the denominator
of the stability function is a product of real linear factors.
Question 5: Prove Butcher's theorem: "There eixtis no 5-stage Runge-Kutta method
of order 5".
Question 6: Show that the 2-step Gauss-Legendre method is of order 4.
**ASSIGNMENT MATERIAL - UNDER CONSTRUCTION :**

Assignment 1 : issued 07.04:30, due 07.05.23, sol. 07.05.30.
From Lambert 1991 (see photocopy of Chapters 1 and 2 distributed in class).
Exercises: 1.6.3, 1.7.2, 1.9.1, 1.10.2, 1.11.1, 2.2.1, 2.4.1, 2.4.2, 2.5.1, 2.5.2.
Assignment 2 : issued 07.05.15, due 07.06.06, sol. 07.06.18.
From Lambert 1991, Chapter 3 .
Exercises: 3.2.1, 3.2.2, 3.3.1, 3.5.1, 3.5.2, 3.81, 3.82.
Hint from Lambert 1973, p. 25, eq. (23) to find the value of t for ex. 3.2.2;
download pdf:
Lambertpg25.jpg
For ex. 3.8.2, use the exact solution for y_1, y'_1, y_2 and y'_2.
You can (a) use the exact value for y_3 and y'_3 and iterate AM once or twice,
or (b) use 4-step AB order 3 predictor to get predicted values y_3 and y'_3 and then
apply AM once,
or (c) solve the implicit AM for y_3, y'_3.
Pick values for h and do several steps.
For h\lambda inside R, the solution should go to zero.
For h too large, the solution should not go to zero.
Download pdf:
Ass2solutions.pdf
Assignment 3 : issued 07.06.06, due 07.06.27, sol. 07.07.03.
See the following pdf for 1 and 2.
1. Redo examples 5.16 and 5.17 from my MAT 2331 or MAT 2784 lecture notes.
2. Exercise 5.3.1. (page 156 in Lambert 1991), Exercise 5.5.1. (page 162 in Lambert 1991).
download pdf:
Ass3.pdf
Assignment 4 : issued 07.06.28, due 07.07.25.
From Lambert 1991.
p. 171 Ex. 5.7.4; pp. 204-205 Ex. 5.12.1, 5.12.3; p. 237 Ex. 6.4.2, 6.4.3;
p. 115 Ex. 4.4.1.
download pdf:
Ass4.pdf
Graded assignments will be returned in class.
Last modified: 2007.07.10