Numerical Methods for Partial Differential Equations

Spectral Methods (446-2)

Winter 2005

Hermann Riecke



  1. Motivation and Introduction
    1. Review of Linear Algebra
  2. Approximation of Functions by Fourier Series
    1. Convergence of Spectral Projection
    2. Approximation of Derivatives
    3. The Gibbs Phenomenon
    4. Discrete Fourier Transformation
    5. Approximation Properties of DFT
    6. Implementation of DFT
  3. Fourier Methods for PDE: Continuous Time
    1. Pseudospectral Method
    2. Galerkin Method
  4. Temporal Discretization
    1. Multistep Schemes
      1. Adams-Bashforth Methods
      2. Adams-Moulton Methods
      3. Leap-Frog
      4. Runge-Kutta Methods
    2. Semi-Implicit Schemes
    3. Integrating-Factor Technique
    4. Operator Splitting
    5. Filtering
  5. Background for Project
  6. Chebyshev Polynomials
    1. Cosine Series
    2. Chebyshev Expansion
    3. Properties of the Chebyshev Polynomials
  7. Chebyshev Approximation
    1. Galerkin Approximation
    2. Pseudospectral Approximation
    3. Derivatives
    4. Pseudospectral Algorithm for Derivatives
      1. Transform Method
      2. Matrix-Multiply Approach
  8. Review of Boundary-Value Problems
    1. Hyperbolic Problems
    2. Parabolic Equations
  9. IBV-Problems: Pseudospectral Methods
    1. Spectra of Modified Differentiation Matrix
    2. Discussion of Time-Stepping Methods for Chebyshev
  10. IBV-Problems: Spectral Methods
    1. Review Fourier Case
    2. Chebyshev Galerkin
    3. Chebyshev tau method
  11. Iterative Mtehods for Implicit Schemes
    1. Simple Iteration
    2. Richardson Iteration
    3. Preconditioning
  12. Higher-Dimensional Problems
  13. 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