# Numerical Methods for Partial Differential Equations

## Spectral Methods (446-2)

### Winter 2005

### Hermann Riecke

- Motivation and Introduction
- Review of Linear Algebra

- Approximation of Functions by
Fourier Series
- Convergence of Spectral Projection
- Approximation of Derivatives
- The Gibbs Phenomenon
- Discrete Fourier Transformation
- Approximation Properties of DFT
- Implementation of DFT

- Fourier Methods for PDE:
Continuous Time
- Pseudospectral Method
- Galerkin Method

- Temporal Discretization
- Multistep Schemes
- Adams-Bashforth Methods
- Adams-Moulton Methods
- Leap-Frog
- Runge-Kutta Methods

- Semi-Implicit Schemes
- Integrating-Factor Technique
- Operator Splitting
- Filtering

- Background for Project
- Chebyshev Polynomials
- Cosine Series
- Chebyshev Expansion
- Properties of the Chebyshev Polynomials

- Chebyshev Approximation
- Galerkin Approximation
- Pseudospectral Approximation
- Derivatives
- Pseudospectral Algorithm for Derivatives
- Transform Method
- Matrix-Multiply Approach

- Review of Boundary-Value Problems
- Hyperbolic Problems
- Parabolic Equations

- IBV-Problems: Pseudospectral Methods
- Spectra of Modified Differentiation Matrix
- Discussion of Time-Stepping Methods for Chebyshev

- IBV-Problems: Spectral Methods
- Review Fourier Case
- Chebyshev Galerkin
- Chebyshev tau method

- Iterative Mtehods for Implicit Schemes
- Simple Iteration
- Richardson Iteration
- Preconditioning

- Higher-Dimensional Problems
- Spectral Methods and Sturm-Liouville Problems

**Assignments:**

HW 1

HW 2

**Resources:**

Preliminary sketch of Lecture Notes

Dave Chopp's Lecture Notes

*Spectral Methods in Matlab* by L.N. Trefethen, SIAM (~$40) (with quite a few matlab programs to download)

Chebyshev and Fourier Spectral Methods by J.P. Boyd (online edition)

**Office Hours:**

TuTh 3:30-4:30 in M458