MA 625: Numerical Methods for Differential Equations Spring, 1998

Section of Professor C. C. Douglas

Class Meets on Tuesdays and Thursdays from 11:00 to 12:15 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.

How to Locate the Instructor

I have 2 offices at UK, which complicates finding me. I spend most of my time on campus in McVey. If you need to leave me a phone message (versus an e-mail message which will get to me quicker), call the McVey number and leave a message with whoever picks up the line.

Building	Room	Phone
McVey	321A	257-2326
Patterson	761	257-6792

Web Usage

The course will use the web extensively. You must know how to use a version 3 (preferably 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

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

I will hand out at most 8 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

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 30th 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

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.

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. 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.

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.

Cheers,
Craig C. Douglas