CPSC 403, Term 2, Winter 2004-2005 Class Home Page

Numerical Solution of Ordinary Differential Equations

Contents of this page

Late Breaking News:

Handouts

# Date Topic
1 Jan. 5 Course Syllabus
2 April 1 Midterm and Solution Set

Homeworks

Collaboration Policy:

You may collaborate with other students in the class on homework questions prior to writing up the version that you will submit. This collaboration may include pseudo-code solutions to programming components. Once you begin writing the version that you will submit, you may no longer collaborate on that question, either by discussing the solution with other students, showing your solution to other students, or looking at the solutions written by other students. You may never share executable code (including Matlab m-files) for homework questions.

You may seek help from the course instructor or TA at any time while preparing your homework solutions. You may not receive help from any other person.

If you feel that you have broken this collaboration policy, you may cite in your homework solution the name of the person from whom you received help, and your grade will be suitably adjusted to take this collaboration into account. If you break this collaboration policy and fail to cite your collaborator, you will be charged with plagarism as outlined in the university calendar. If you have any questions, please contact the instructor.

More information is available from (among other sources):

Homework Submission Policy:

Homeworks:

# Topic Assigned Due Date Assignment Solution Other Files
1 ODE Property Review January 10 January 24 Problem Set Solutions Source code for example of how to solve question 5a (for a different ODE). Example output in postscript, jpeg, or png formats.
2 Basic Methods January 25 February 7 Problem Set Solutions none.
3 Runge-Kutta Single Step Methods February 7 February 28 Problem Set Solutions none.
4 Linear Multistep Methods & Boundary Value Problems February 28 March 21 Problem Set Solutions A driver routine to generate the convergence plot for question 3: converge.m. Implementation of fixed timestep Adams-Bashforth: adamsBashforth.m. Table of BDF alpha coefficients, order 1-8
5 Boundary Value Problems & Differential Algebraic Equations March 21 April 8 Problem Set Solutions none.

From the Calendar

Investigation of practical computational methods for ordinary differential equations. Multistep and Runge-Kutta methods for initial value problems. Control of error and stepsize. Special methods for stiff equations. Shooting, finite difference and variational methods for linear and nonlinear boundary value problems.

Course Overview

This course will investigate practical computational techniques for ordinary differential equations. The need to simulate solutions of such equations, and more generally of continuous dynamical systems, arises in many application. Topics include:

Course Details

Intended Audience: Senior undergraduates and junior graduate students in:

Prerequisites:

Instructor: Ian Mitchell

TA: Dan Li

Course Communication:

Lectures: 14:00 - 15:00, Monday / Wednesday / Friday, Dempster 301

Grades: Your final grade will be based on a combination of:

To pass the course, you must obtain a 50% overall mark and pass the final exam. The final exam will cover material from the entire course.

Textbook:

Other References:

The Schedule:

# Date Topic Links Required Readings Optional Readings
1 Jan 5 Course syllabus and administration. Collaboration guidelines. Professor introduction. Ian Mitchell's homepage none AP Preface
2 Jan 7 Types and Applications of Ordinary Differential Equations. none AP 1 H 9.1
3 Jan 10 Well posed IVPs. Problem Stability. Scalar Test Equation. John Polking's pplane software for phase plane plots of ODEs. AP 2.1 BF 5.1, 5.9; H 9.2
4 Jan 12 Matrix Test Equation. Forward Euler (FE). none AP 2.2, 2.4 AP 2.3, 2.5
5 Jan 14 FE. Convergence, Accuracy, Consistency and 0-Stability. none AP 3.1-3.2 BF 5.2, 5.10; H 9.3.1-9.3.2
6 Jan 17 Absolute Stability. Backward Euler (BE). Matlab's Introduction to initial value ODEs AP 3.3-3.4 BF 5.10; H 9.3.3
7 Jan 19 Stiff ODEs. Stiff Decay. Rough Problems. m-file for the sawtooth integration. AP 3.5, 3.7 BF 5.11; H 9.3.4
8 Jan 21 Trapezoidal Method. Taylor Series Methods. Runge-Kutta Methods (RK Methods). none AP 3.6, 4.0-4.1 BF 5.3, 5.4; H 9.3.5-9.3.6
9 Jan 24 Properties of Runge-Kutta Methods none AP 4.2-4.4 BF 5.4; H 9.3.6
10 Jan 26 Principle Error Function & Global Error ODE. Error Tolerance. Error Control. none AP 4.5 none
11 Jan 28 Error Estimation. Solution Sensitivity. m-file for sensitivity analysis of example 4.7 (the toy car). AP 4.6 BF 5.5
12 Jan 31 Sensitivity. Implicit Runge-Kutta. none AP 4.7 none
13 Feb 2 Linear Multistep Methods (LMMs). Consistency & Order. none AP 5.0, 5.2.1 BF 5.6; H 9.3.8
14 Feb 4 Examples of LMMs: Adams-Bashforth, Adams-Moulton, BDFs. Initial Conditions. none AP 5.1 none
15 Feb 7 LMM 0-Stability, Convergence & Absolute Stability. none AP 5.2.2, 5.2.3, 5.3 none
16 Feb 9 Implicit Schemes. Functional Iteration. Predictor-Corrector. Newton & Modified Newton. none AP 3.4.2, 5.4 AP 5.5; BF 2.2-2.3, 10.1-10.2; H 5.5.2-5.5.3, 5.6.1-5.6.2
17 Feb 11 A Example of Failure in Matlab ODE45. m-file for the coupled oscillator example. none Analysis Still Matters..." by Skufca.
Feb 14-18 Midterm break (no class).
18 Feb 21 Properties of Boundary Value Problems. none AP 6.0 H 10.1
19 Feb 23 Linear BVPs. Stability of BVPs. none AP 6.1-6.2 H 10.2
20 Feb 25 BVP Dichotomy. Isolated Solutions. none AP 6.2 none
21 Feb 28 BVP Stiffness. Reformulation Tricks. Simple Shooting. none AP 6.3-6.4, 7.1 BF 11.1-11.2; H 10.3
22 March 2 Problems with Simple Shooting none AP 7.1.1 none
23 March 4 Multiple Shooting none AP 7.2 none
March 7 Midterm Exam
24 March 9 Guest Lecturer Robert Bridson: Methods for Second Order ODEs.
25 March 11 Guest Lecturer Robert Bridson: Symplectic Methods.
26 March 14 Linear Multiple Shooting Formulation. Introduction to Finite Differences none AP 8.0 none
27 March 16 Midpoint Method for Finite Difference. none AP 8.1.1 BF 11.3-11.4; H 10.4
28 March 18 Finite Difference Scheme Properties. none AP 8.1.2 AP 8.2
29 March 21 Higher Order Schemes for BVPs: Extrapolation, Deferred Correction, Runge-Kutta Style Collocation. none AP 8.3 BF 4.2, 11.3-11.4; H 8.7
30 March 23 Higher Order Schemes for BVPs: General Collocation, including Spectral Methods none none H 10.5
March 25 Good Friday Holiday (no class).
March 28 Easter Monday Holiday (no class).
31 March 30 Finite Element Methods: Least Squares, Galerkin and Ritz. none none BF 11.5; H 10.6
32 April 1 Error Estimation & Mesh Generation. Damped Newton Method none AP 8.4.1, 8.5 AP 8.4; BF 10.5
33 April 4 Initial Guess Generation by Continuation. Differential Algebraic Equations. none AP 8.4.3, 9.0 AP 8.8; BF 10.5
34 April 6 DAE Index. Solving DAEs by Direct Discretization. none AP 9.1, 10.1 AP 9.3-9.4, 10.3
35 April 8 Solving DAEs by Stabilization. Course Review. none AP 9.2, 10.2 none
April 20 Final Exam. 12:00 noon. CEME 1212.

Links:

CPSC 403 Term 2 Winter 2004-2005 Class Page
maintained by Ian Mitchell