By John P. Boyd
Thoroughly revised textual content makes a speciality of use of spectral the right way to remedy boundary worth, eigenvalue, and time-dependent difficulties, but in addition covers Hermite, Laguerre, rational Chebyshev, sinc, and round harmonic services, in addition to cardinal services, linear eigenvalue difficulties, matrix-solving tools, coordinate variations, round and cylindrical geometry, and extra. comprises 7 appendices and over one hundred sixty textual content figures.
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Extra resources for Chebyshev and Fourier Spectral Methods, Second Edition
It is for this reason that our series, instead of being the more familiar Taylor expansions, will be Fourier or Chebyshev or Legendre series instead. If its disk of convergence is too small to include the boundaries, the power series will give nonsense. In contrast, the success of spectral methods is guaranteed as long as the target interval, x ∈ [−1, 1], is free of singularities. 8 Stalking the Wild Singularity or Where the Branch Points Are It is sometimes possible to identify the type and nature of the singularities of the solution to a differential equation a priori without knowing the solution.
6 Darboux’s Principle Theorem 1 (DARBOUX’S PRINCIPLE: SINGULARITIES & CONVERGENCE) For all types of spectral expansions (and for ordinary power series), both the DOMAIN of CONVER GENCE in the complex plane and also the RATE of CONVERGENCE are controlled by the LOCATION and STRENGTH of the GRAVEST SINGULARITY in the complex plane. “Singularity” in this context denotes poles, fractional powers, logarithms and other branch points, and discontinuities of f (z) or any of its derivatives. Each such singularity gives its own additive contribution to the coefficients an in the asymptotic limit n → ∞.
Historically, the “non–interpolating” methods were developed first. For this reason, the label “spectral” is sometimes used in a narrow sense as a collective tag for the “non– interpolating” methods. In these notes, we shall use “spectral” only as catch–all for global expansion methods in general, but the reader should be aware of its other, narrower usage. ). Many spectral models of time-dependent hydrodynamics split the calculation into several subproblems and apply different techniques to different subproblems.