Numerical Methods for Partial Differential Equations

Spectral Methods (446-2)

Spring 2007

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 Template of Matlab Code

HW 3
Have a look at some interesting journal papers related to this homework
Ouyang and Swinney, Transition from a uniform state to hexagonal and striped Turing patterns
Nature 352 (1991) 610
Bodenschatz et al, Transitions between patterns in thermal convection
Phys. Rev. Lett. 67 (1991) 3078
Harrison et al. Mechanisms of ordering in striped patterns
Science 290 (2000) 1558

HW 4

HW 5
Have a look at:
H. Chate, Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation
Nonlinearity 7 (1994) 185
Aranson and Kramer, The world of the complex Ginzburg-Landau equation
Rev. Mod. Phys. 74 (2002) 99

HW 6 improved movie call
Have a look at:
H. Levine, D.A. Kessler, W.-J. Rappel Directional sensing in eukaryotic chemotaxis: A balanced inactivation model PNAS 103 (2006 9761


Resources:
Sketch of Lecture Notes
Sketch of Lecture Notes wide margin
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:
Mo 1-2, We 4-5, Fr 1-2 in M458