Special Topics in Numerical Analysis:
Spectral Methods

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Course Description

Various advanced topics in numerical analysis and scientific computing, such as iterative methods, eigenvalue computations, approximation theory, finite element methods, inverse problems, nonlinear conservation laws, computational fluid dynamics.

Prerequisites

Familiarity with numerical analysis at the level of AMath 584-5-6.

Syllabus

We will work through the book Spectral Methods in Matlab by L. N. Trefethen (SIAM, 1999). This gives a nice introduction to the use of spectral methods for obtaining highly accurate numerical solutions to differential equations.

The chapters in this book are:

• Differentiation Matrices
• Unbounded Grids: The Semidiscrete Fourier Transform
• Periodic Grids: The DFT and FFT
• Smoothness and Spectral Accuracy
• Chebyshev Differentiation Matrices
• Boundary Value Problems
• Chebyshev Series and the FFT
• Eigenvalues and Pseudospectra
• Time-Stepping and Stability Regions
• Polar Coordinates
• Integrals and Quadrature Formulas
• More about Boundary Conditions
• Fourth-Order Problems

Reading and Homework Assignments

• For Wednesday 4/3: Read Chapter 1 and do all exercises.
• For Monday 4/8: Read Chapter 2 and do all exercises.
• For Wednesday 4/10: Read Chapter 3 and start these exercises.
• For M 4/15 and W 4/17: Finish Chapter 3 and do Chapter 4.

Other Links

• Here is a list of some Matlab resources available on the net:
  • List of Matlab Resources at Mathworks site
  • Matlab Hypertext Reference, Portland State University
  • Matlab Quick Reference Guide, University of Waterloo