No Title

**Numerical Solution of Partial Differential Equations**

**Spectral Methods**

##### ESAM D-46-2

Hermann Riecke

Newsgroup for discussions/questions

Homework Assignments:

HW 1
Project 1
HW 2
Project 2

In this class spectral methods will be discussed for the solution of partial
differential equations. In two projects the students will implement
Fourier- and Chebyshev-spectral methods and will investigate their properties.
The case study for the Fourier method will employ the nonlinear
Schrödinger/Ginzburg-Landau equation, which describes, for instance, phase transitions, soliton pulses in fiber optics, weakly nonlinear waves in general,... In the Chebyshev
project a set of equations will be solved which describe oscillatory and pattern-forming
chemical reactions. In both projects graphical animation will aid the developing of the
codes as well as the interpretation of the results. In particular the
Fourier-spectral code will enable the students to solve efficiently a large variety of
other partial differential equations without too much modification.

The technical topics to be discussed will include:

Approximation properties of Fourier and Chebyshev series,
pseudo-spectral vs. Galerkin methods, review of time-stepping methods (stability),
initial-boundary-value problems, Chebyshev-tau method, iterative methods (implicit schemes,
preconditioning), higher-dimensional problems.

Strongly recommended book:

C. Canuto et al., *Spectral Methods in Fluid Dynamics*, Springer.

Time and Place: Tu Th 2:30-4:00 in A110 (computer projection room)

Do you have any questions?

Phone: 491-8316, e-mail: h-riecke@nwu.edu

* Hermann Riecke *

Fri Jan 16 1997