AMATH 571
SLN 1219, MW 3:30-4:50, Loew Hall 111
(Prerequisites: AMATH 585-586)

Spectral Methods

Instructor:

Professor Chris Bretherton
ATG 710
tel: 685-7414
fax: 543-0308
breth@atmos.washington.edu
office hours: Tu F 1:30-2:30 or by appt.

Schedule

Handouts

Homework

Matlab scripts

Grades

Message Board

Catalog Description

Analysis and application of spectral methods for the numerical solution of differential equations. Fourier methods and the FFT; collocation methods; polynomial interpolation and Chebyshev series; approximation theory and spectral accuracy; boundary conditions.

Learning Objectives and Instructor Expectations

Our goal is to provide a research-level intellectual foundation for numerical solution of PDEs using spectral methods and for debugging and analyzing the results. The focus will be on applications in nonlinear waves and fluid dynamics, implementing and testing algorithms in Matlab.

Prerequisite: Amath 584, 585, 586, and a good working knowledge of Matlab.

Textbook

Required text (order on line or through the University Book Store):

Trefethen, N.,2000: Spectral methods in Matlab, SIAM Press (paperback).

The unit on wavelets will draw on lecture notes (to be handed out) and the Matlab Wavelet Toolbox Users Guide (1997, MathWorks).

Syllabus

Advanced applications of spectral methods (8 weeks),

Initial value problems using Fourier spectral methods. Application to nonlinear waves and 2D fluid 'turbulence'.
How does the Fast Fourier Transform work?
Chebyshev-based methods, including boundary conditions.
Spectral algorithms for numerical interpolation and quadrature.
Spectral methods on a sphere (if we have time and interest).

(3)Wavelet analysis. (2 weeks - again if there is interest)

Haar discrete wavelet transform
Multiresolution analysis and wavelet-based compression.
Other wavelets.
Continuous Wavelet Transform.

Schedule

Date	Occasion
Wed 4 Jan	First day of classes
Mon 16 Jan	No class (MLK Day)
Mon 30 Jan-Thu 2 Feb	No class or office hours (Instructor at meeting in Atlanta)
Mon 20 Feb	No class (President's Day)
Wed 8 Mar	Last day of classes

Handouts

Jan 4: The Discrete Fourier Transform.
Jan 9: Fourier spectral methods for Poisson's equation
Jan 11: Fourier spectral simulation of 2D fluid flow
Jan 18: Fourier cardinal functions and differentiation matrices: lecture notes and overheads.
Jan 25 - Feb 6: Chebyshev spectral methods --lecture notes and overheads.
Feb. 8-13: Quadrature using Chebyshev and Legendre polynomials --lecture notes and overheads.
Feb. 15-22: Chebyshev spectral methods for eigenvalue problems
Feb. 27: Chebyshev spectral methods for fourth-order BVPs
Mar. 1-8: Wavelet analysis with the Matlab wavelet toolbox.

Homework

Homework	Due Date	Solutions
Homework 1	Due Fr 27 Jan	HW1 solutions
Homework 2	Due Mon 6 Feb	HW2 solutions
Homework 3	Due Wed 15 Feb	HW3 solutions
Homework 4	Due Wed 1 Mar	HW4 solutions
Final assignment	Due Th 16 Mar, 5 p.m.	Solutions

Grading

Your course grade will be based on your homework (50%), take-home midterm and final projects (20% each), and class participation (10%, based on in-class discussion participation and attendance). There will be no in-class exams. Matlab-based homework will be assigned quasi-weekly. You are encouraged to discuss homework solution strategy and results with your classmates but not more senior students. However, I ask that each of you compose and write your own Matlab scripts and write up your own discussion of your results. No collaboration will be allowed on the midterm and final projects.

<breth@atmos.washington.edu>