By John P. Boyd
Thoroughly revised textual content makes a speciality of use of spectral the way to resolve boundary worth, eigenvalue, and time-dependent difficulties, but additionally covers Hermite, Laguerre, rational Chebyshev, sinc, and round harmonic capabilities, in addition to cardinal features, linear eigenvalue difficulties, matrix-solving equipment, coordinate ameliorations, round and cylindrical geometry, and extra. comprises 7 appendices and over a hundred and sixty textual content figures.
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Extra info for Chebyshev and Fourier Spectral Methods, Second Edition
2: the smoother f (x), the faster the convergence of its spectral series. ) and the asymptotic spectral coefficients. Darboux’s Principle implies a corollary that clarifies the Method of Model Functions. Corollary 1 (SINGULARITY-MATCHING) If two functions f (z) and g(z) have convergence-limiting singularities of the same type and strength, then their asymptotic spectral coefficients are the same in the limit n → ∞. If f (z) − g(z) is singular, but more weakly singular than either f (z) or g(z), the difference between the spectral coefficients of f (z) and g(z) decreases as an algebraic function of the degree n.
7 Convergence Domains: Why Taylor Series Don’t Work For all the standard basis functions, the spectral series converges in a domain of known shape in the complex plane. Since the singularities of f (x) control the asymptotic form of the spectral coefficients (Darboux’s Principle), it follows that the size of the convergence domain and the rate of convergence at a given point within the domain are both controlled by the singularities of f (x), too. Theorem 2 (CONVERGENCE DOMAIN in COMPLEX PLANE) Barring a few exceptional cases, a spectral series converges in the largest domain in the x-plane, of a certain basis-dependent shape, which is free of singularities.
The third keystone is that from Darboux’s Principle, and limited knowledge about a function, such as whether it is or is not pathological on the solution domain, we can predict rates of convergence for spectral series and spectral approximations to differential equations. Several qualitatively different rates are possible: algebraic, geometric, subgeometric, and supergeometric. The fourth keystone is that from model functions and Darboux’s Principle, we can develop some rules-of-thumb that allow us to qualitatively estimate a priori how many degrees of freedom N are needed to resolve a given physical phenomenon.