Synopsis for Spectral Methods for ODEs and PDEs

Spectral methods are a standard tool for solving ODEs and PDEs numerically through the use of Fourier and Chebshev expansions and interpolants. They are noted for their elegance and high accuracy and are based on fundamental mathematics useful to any applied mathematician. This course will be a hands-on introduction to spectral collocation algorithms, including theoretical fundamentals but with greater emphasis on practical implementation in Matlab, following the instructor's short textbook Spectral Methods in Matlab.

Syllabus

1. Differentiation matrices
2. Unbounded grids: the semidiscrete Fourier transform
3. Periodic grids: the DFT and FFT
4. Smoothness and spectral accuracy
5. Polynomial interpolation and clustered grids
6. Chebyshev differentiation matrices
7. Boundary value problems
8. Chebyshev series and the FFT
9. Eigenvalues and pseudospectra
10. Time-stepping and stability regions
11. Polar coordinates
12. Integrals and Quadrature formulas

Reading List

1. L. N. Trefethen, Spectral Methods in Matlab, SIAM 2000