Detailed Course Outline

• Introduction and Preliminaries ( Lecture 1)

  • Historical background

  • Some examples of spectral methods:
    • For Poisson equation in 1D
    • For heat equation in 1D
    • For wave equation in 1D

  • Orthogonal polynomials
    • Sturm-Liouville problems
    • The continuous Fourier expansion
    • Legendre polynomials
    • Chebyshev polynomials
    • Jacobi polynomials
    • Error estimates of polynomial approximations

  • Review of time discretization methods

  • Review on iterative methods and preconditioning

  ---

• Spectral-Collocation Methods
  • Introduction

  • Differentiation matrices for Fourier collocation methods

  • Differentiation matrices for polynomial basis functions

  • Chebyshev collocation methods

  • Collocation method in the weak form and preconditioning

  ---

• Spectral-Galerkin Methods
  • Introduction

  • General setup

  • Fourier spectral and pseudospectral methods

  • Legendre-Galerkin method

  • Chebyshev-Galerkin method

  • Spectral-Galerkin methods for higher-order equations

  • Error estimates

  ---

• Spectral Methods in Unbounded Domains
  • Introduction

  • Hermite spectral methods

  • Laguerre spectral methods

  • Spectral methods using rational functions

  • Error estimates

  ---

• Applications
  • Introduction

  • In fluid dynamics

  • In heat transfer

  • In materials sciences

  • In quantum physics and nonlinear optics

  • In plasma and particle physics

  • In biology

  ---