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Spectral Methods in MATLAB
TLDR
This paper presents a meta-analyses of Chebyshev differentiation matrices using the DFT and FFT as a guide to solving fourth-order grid problems.
Hydrodynamic Stability Without Eigenvalues
TLDR
A reconciliation of findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically.
Fourth-Order Time-Stepping for Stiff PDEs
A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as
Barycentric Lagrange Interpolation
Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.
Spectra and pseudospectra : the behavior of nonnormal matrices and operators
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Spectra and Pseudospectra
TLDR
Aerodynamics provided the impetus for some significant work on the theory and computation of matrix eigenvalues, and numerical methods for solving eigenproblems were not available in those early days.
Is Gauss Quadrature Better than Clenshaw-Curtis?
TLDR
Comparisons of the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis are compared, and experiments show that the supposed factor-of-2 advantage of Gaussian quadratures is rarely realized.
Spectra and pseudospectra
The five sections of these notes will one day be the first five chapters of a book, to appear some time after 2001.
Parabolic and hyperbolic contours for computing the Bromwich integral
TLDR
This work analyzes two recently proposed contours, namely a parabola and a hyperbola, for the numerical inversion of the Laplace transform and determines estimates for the optimal parameters that dene these contours.
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