AMATH 585
http://www.amath.washington.edu/courses/585-winter-2002/
Approximate and Numerical Analysis II
Instructor:
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Professor
Chris Bretherton
ATG 704
tel: 685-7414
fax: 685-9302
breth@atmos.washington.edu
office hours: MW 3:30-4:20, Th 9:30-10:20 or by appointment
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Course Description
Numerical methods for steady-state differential equations. Two-point
boundary value problems and elliptic equations. Finite difference, spectral
and finite element methods. Numerical interpolation, integration,
application of discrete Fourier transform.
Prerequisites
AMATH 581 or 584. An acceptable substitute, by permission of the instructor,
is an undergraduate course in numerical analysis, including numerical linear
algebra and ordinary differential equations, along with fluency in MATLAB (or
willingness to learn it quickly.)
Syllabus
- Numerical interpolation and integration (1+ lecture)
- Polynomial interpolation
- Splines
- Applications of discrete Fourier transform (2+ lectures)
- Properties of DFT
- Numerical differentiation
- Two-point boundary value problems for ODEs (5 weeks)
- Character of solution; boundary conditions.
- Finite difference method for linear problem u'' = f, BCs at x = 0,1.
- Accuracy, convergence, stability.
- Finite difference methods for nonlinear problem
- Shooting method
- Boundary layers and nonuniform grids
- Function space methods
- Galerkin (spectral) methods.
- Finite element method.
- Multidimensional boundary value problems for PDEs (4 weeks)
- Laplace/Poisson equation - some physical examples
- Finite difference method; solution via Gaussian elimination.
- Rectangular domains
- Spectral methods for homogeneous BCs
- DFT-based solution to finite difference equations.
- Finite Element Method and Matlab PDE toolbox
- Iterative solution of sparse matrix problems
- Jacobi, Gauss-Seidel, SOR
- Multigrid
- Conjugate gradient and preconditioning
Recommended Texts
There are no assigned
textbooks for this class. Some of the material is covered by course notes
written by Prof. Randy LeVeque, which you may download
as a pdf file and print. These will be supplemented by other handouts.
Grading
The entire course grade is based on homework assigned every week or two, which
will involve extensive use
of Matlab. You are welcome to consult with your classmates
about how to do the homework, but I ask that everyone write all their own
code. You will be asked to do one homework in the middle of the term and one
at the term's end without help from others (as take-home 'exams'), although
you certainly may consult the course notes or other books.
You may view your homework
grades on-line.
Before doing so for the first time, you must
request a password.
Important note: You will be asked for your student number when you
request a password. Do not include any leading zeros when you type your
student number, or else your request will be rejected. For example, if your
student number were 0012345, you'd type 12345. If it were 0123456, you'd type 123456, and if it were 1234567 you'd type 1234567.
Special days
- We 9 Jan, We 13 Mar, Fr 15 Mar - I will be out of town at
meetings.
- These classes rescheduled into six half hour
class extensions (3:20-3:50 pm) on the following Mondays:
- Mo 14 Jan
- We 30 Jan
- We 6 Feb
- We 13 Feb
- We 20 Feb
- We 27 Mar
- Vacations - no class
- Mo 21 Jan: MLK Day
- Mo 18 Feb: Presidents Day - no class
- We 20 Mar: Final assignment due by 5 pm, Bretherton's mailbox,
ATG410.
.