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 ...