MA 625: Numerical Methods for Differential Equations Spring, 1999

Section of Professor C. C. Douglas

Class Meets on Tuesdays and Thursdays from 14:00 to 15:15 in McVey 327

Graduate Handbook Description: MA 625, Numerical Methods for Differential Equations

Numerical Methods for Two-point Boundary Value Problems.

Finite difference methods for linear and nonlinear second order equations: Richardson extrapolation, deferred corrections, Numerov's method. First order systems. Finite-element methods: a) Rayleigh-Ritz Galerkin methods for second order problems: piecewise polynomials, B-splines, b) Spline collocation at Gaussian nodes. Shooting methods: review of methods for solving initial value problems, ordinary shooting, multiple shooting.

Numerical Methods for Parabolic Initial/Boundary Value Problems.

Basic finite difference methods for a) the heat equation; b) problems with variable coefficients; c) nonlinear problems. Finite element methods: Galerkin or spline collocation in space and finite difference discretizations in time.

Prerequisite: MA/CS/EGR 537 or consent of instructor.

This course will give the students a solid foundation in solving differential equations both theoretically and computationally. Algorithms to solve problems will be emphasized. Theory and applications will be equally weighted. At the end of the course the students should know what type of algorithm to try to solve a problem, why it works, and how well it should have worked.

Shooting methods and collocation will not be emphasized. More modern methods for solving PDE's, e.g., multigrid and domain decomposition, will be substituted.

Office Hours

My primary office is 321A McVey Hall (I have another office in Patterson). If you need to leave me a phone message (versus sending me email at douglas@ccs.uky.edu, which will get to me quicker), call 257-2326 and leave a message with either Zetta Vaught or Sandy Leachman.

Never, ever walk across campus to one of my offices without calling first. Feel free to drop in without an appointment on Wednesdays and Thursdays from 8:15-9:30 after locating me. I am also available for appointments. When in doubt, call first.

Web Usage

The course will use the web extensively. You must know how to use a version 4 type browser like ones provided by Netscape or Microsoft. The syllabus can be found as a link in the class home page. It is located at the URL

http://www.ccs.uky.edu/~douglas/ma625

Please bookmark this URL and check it often. Homework will be posted through the web pages in this folder. The class web page has a number of hyperlinks that you will find either useful or essential.

Homework, the Late Policy, Exams, Grading, and Dead Sources

I will hand out homework assignments during the semester. The pencil and paper parts should be turned in at the beginning of class on the due date. Codes should be e-mailed before class to

douglas@ccs.uky.edu

Some assignments may take longer than others and be called projects. Please check the course home page for the homework weightings.

I will take late homework only if there is a compelling reason; please contact me in advance, if possible. I will give you an extension for serious health problems, job interviews, death of a relative, or a similar, serious situation. Do not come and tell me that so-and-so's course is more important than mine and you did their assignment or project instead of mine.

Grading will be very simple. Since this is a graduate level course, +'s and -'s will not be given. The homework will count 100% of the grade. There will be no exams in this course. You are free on Thursday, April 29th to enjoy life after MA 625.

All UK faculty are required to state in the syllabus the grading system. The system can change if I give you adequate warning. As a rule, having the following percent of the scaled points will earn a grade of

Grade	Minimum %
A	80
B	65
C	50

If you are caught cheating, you will automatically get an E and all sorts of academic and possibly legal problems will arise.

Textbook

K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994, ISBN 0-521-42922-6.

When a lecture comes from another book, there will be a class handout and references will be given.

Other Books of Interest

G. E. Forsythe and W. R. Wasow, Finite-Difference Methods for Partial Differential Equations, John Wiley & Sons, New York, NY, 1960.

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, UK, 1990.

L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley & Sons, New York, NY, 1982.

John H. Mathews, Numerical Methods for Mathematics, Science and Engineering, Prentice-Hall, Englewood Cliffs, NJ (USA), 1992, ISBN 0-13-624990-6.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, Wiley-Interscience, New York, 1967 and 1994 (reprint).

M. H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1973.

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Chapman and Hall, London, 1989.

Y. W. Kwon and H. Bang, The Finite Element Method Using MATLAB, CRC Press, Boca Raton, FL, 1997.

R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.

Outline of the Course

As many of the following topics will be covered as time permits.

1. Introduction

2. Ordinary Differential Equations

• Model problems
• One step methods
• k-th order Methods
• Multistep methods
• Shooting methods

3. Parabolic Equations in 1-D

• A simple model problem
• Fourier series approximation
• Difference schemes
• Convergence
• The maximum principal
• More general model problems
• Polar coordinates
• Mildly nonlinear equations

4. Parabolic Equations in 2-D and 3-D

• A simple model problem in a rectangular domain
• Alternating Direction Implicit(ADI) methods in 2-D
• Operating splitting
• General domains
• General model problems

5. Hyperbolic Equations (primarily 1-D)

• A simple model problem
• Characteristics
• Cauchy-Friedrichs-Lewy (CFL) condition
• Upwind schemes & analysis
• Lax-Wendroff scheme
• Phase and amplitutde errors
• > 1-D problems

6. Consistency, stability, and convergence

• Finite differences
• Lax equivalence theorem
• Calculating stability conditions
• Conservation laws

7. Linear Elliptic Equations in 2-D

• Laplace's equation
• General diffusion equations
• Curved domains and boundary conditions
• Maximum principaland error analysis
• Variational formulations (finite elements/volumes)

8. Multigrid and Domain Decomposition

• Model problems
• Algorithms
• Convergence rates
• Cost

9. Parallel Computing

• Bottlenecks
• Overlapping computation and I/O
• Domain decomposition + multigrid versus
  multigrid + domain decomposition

Companion Software

Numerical analysts need to know Fortran in order to read old codes and re-use them. Translators (e.g., f2c or c++2j) produce sufficiently bad code as to make the translation unusable from a wall clock point of view. This does not mean that all numerical analysts should program only in Fortran. Many applications are better suited to Ada, Matlab, Lisp, C, C++, or Java.

This course will use Matlab whenever a programming assignment is necessary unless stated otherwise. If you have a Windows 95/NT based PC, you might consider purchasing the student edition of Matlab from Prentice-Hall. There are Matlab software keys at UK for all of the Windows NT machines in the classroom building, there are many keys at engineering division computing clusters, computer science, and a few in the math department.

Note that for students to fully appreciate the parallel computing part of this course, it will be necessary for the students to get an account on the HP Exemplar at the computing center. The instructor will sponsor the account if necessary. Note that parallel programming cannot be done using Matlab, but will require using Fortran, C, or C++.

Cheers,
Craig C. Douglas