Chebyshev & Fourier Spectral MethodsThe goal of this book is to teach spectral methods for solving boundary value, eigenvalue, and time-dependent problems. Although the title speaks only of Chebyshev polynomials and trigonometric functions, the book also discusses Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions. These notes evolved from a course I have taught the past five years to an audience drawn from half a dozen different disciplines at the University of Michigan: aerospace engineering, meteorology, physical oceanography, mechanical engineering, naval architecture, and nuclear engineering. With such a diverse audience, this book is not focused on a particular discipline, but rather upon solving differential equations in general. The style is not lemma-theorem-Sobolev space, but algorithms guidelines-rules-of-thumb. Although the course is aimed at graduate students, the required background is limited. It helps if the reader has taken an elementary course in computer methods and also has been exposed to Fourier series and complex variables at the undergraduate level. However, even this background is not absolutely necessary. Chapters 2 to 5 are a self contained treatment of basic convergence and interpolation theory. |
Contents
Introduction 1 Series Expansions | 1 |
Choice of Basis Functions | 2 |
Comparison with Finite Element Methods | 3 |
Copyright | |
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Common terms and phrases
accuracy algebraic algorithm aliasing analytical antisymmetric apply approximation asymptotic basis functions basis set boundary conditions boundary layers boundary value problem Boyd branch points calculation Canuto cardinal functions Chapter Chebyshev functions Chebyshev polynomials Chebyshev series collocation compute coordinate cos(x Crank-Nicholson decay defined degree denote derivatives diagonal differential equation diffusion dimensions domain eigenfunctions endpoints error evaluate exact solution example expansion exponential factor finite difference methods flow Fourier series fractional step Galerkin's method Gaussian elimination Gegenbauer polynomials Hermite functions infinite inner product integration interpolation points interval Legendre linear mapping Newton's Newton's iteration nonlinear numerical one-dimensional operations Orszag orthogonal parameter parity poles power series pre-conditioning pseudospectral method rational Chebyshev functions residual Richardson's iteration scheme second order shows sin(x sine singularities solve spectral coefficients spectral methods spherical harmonics splitting symmetry theorem transform trigonometric truncation two-dimensional velocity wave wavenumber zero