Книга: Spectral method: Applied Mathematics, Computational Science, Partial Differential Equation, Fast Fourier Transform, Ordinary Differential Equation, Chebyshev Polynomials, Finite Element Method

High Quality Content by WIKIPEDIA articles! Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain partial differential equations (PDEs), often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called 'exponential convergence' being the fastest possible. PDEs describe a wide array of physical processes such as heat conduction, fluid flow, and sound propagation. In many such equations, there are underlying 'basic waves' that can be used to give efficient algorithms for computing solutions to these PDEs. In a typical case, spectral methods take advantage of this fact by writing the solution as its Fourier series, substituting this series into the PDE to get a system of ordinary differential equations (ODEs) in the time-dependent coefficients of the trigonometric terms in the series (written in complex exponential form), and using a time-stepping method to solve those ODEs.