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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:

Hermann Riecke
Fri Jan 16 1997