# Numerical Methods for Partial Differential Equations

## Spectral Methods (446-2)

### Spring 2007

### 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 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