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TIME DEPENDENT PROBLEMS
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SPECTRAL METHODS FOR
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SPECTRAL METHODS FOR
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 ...
Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998