January 22, 1998

Abstract:

We review several topics concerning spectral approximations of time-dependent problems, primarily -- the accuracy and stability of Fourier and Chebyshev methods for the approximate solutions of hyperbolic systems.

To make these notes self contained, we begin with a very brief overview of Cauchy problems. Thus, the main focus of the first part is on hyperbolic systems which are dealt with two (related) tools: the energy method and Fourier analysis.

The second part deals with spectral approximations. Here we introduce the main ingredients of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, differentiation matrices ...

The third part is devoted to Fourier method for the approximate solution of periodic systems. The questions of stability and convergence are answered by combining ideas from the first two sections. In this context we highlight the role of aliasing and smoothing; in particular, we explain how the lack of resolution might excite small scales weak instability, which is avoided by high modes smoothing.

The forth and final part deals with non-periodic problems. We study the stability of the Chebyshev method, paying special attention to the intricate issue of the CFL stability restriction on the permitted time-step.

LECTURE NOTES DELIVERED AT
ECOLE DES ONDES
``Méthodes numériques d'ordre élevé
pour les ondes en régime transitoire''
Inria - Rocquencourt, January 24-28 1994

 
• Contents
• TIME DEPENDENT PROBLEMS
  • Initial Value Problems of Hyperbolic Type
    • The wave equation -- hyperbolicity by the energy method
    • The wave equation -- hyperbolicity by Fourier analysis
    • Hyperbolic systems with constant coefficients
    • Hyperbolic systems with variable coefficients
  • Initial Value Problems of Parabolic Type
    • The heat equation -- Fourier analysis and the energy method
    • Parabolic systems
  • Well-Posed Time-Dependent Problems
• SPECTRAL APPROXIMATIONS
  • The Periodic Problem -- The Fourier Expansion
    • Spectral accuracy
  • The Periodic Problem -- The Fourier Interpolant
    • Aliasing and spectral accuracy
    • Fourier differentiation matrix
    • Fourier interpolant revisited on an even number of gridpoints
  • The (Pseudo)Spectral Fourier Expansions - Exponential Accuracy
  • The Non-Periodic Problem -- The Chebyshev Expansion
    • Spectral accuracy
  • The Non-Periodic Problem -- The Chebyshev Interpolant
    • Chebyshev interpolant at Gauss gridpoints
    • Chebyshev interpolant at Gauss-Lobatto gridpoints
    • Exponential convergence of Chebyshev expansions
    • Chebyshev differentiation matrix
• THE FOURIER METHOD
  • The Spectral Fourier Approximation
    • Stability and convergence
  • The Pseudospectral Fourier Approximation
    • Is the pseudospectral approximation with variable coefficients stable?
  • Aliasing, Resolution and (weak) Stability
    • Weighted -stability
    • Algebraic stability and weak -instability
    • Epilogue
  • Skew-Symmetric Differencing
  • Smoothing
• THE CHEBYSHEV METHOD
  • Forward Euler -- the CFL Condition
    • Problems with inhomogeneous initial-boundary conditions
  • Multi-level and Runge-Kutta Time Differencing
  • Scalar Equations with Variable Coefficients
• About this document ...