Spectral Methods for Time-Dependent Problems

By: Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb

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Hardcover | 11 January 2007

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284 Pages

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Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Industry Reviews

'The book is excellent and will be valuable for post-graduate students, researchers and scientists working in applied sciences and mainly in the numerical analysis of time-dependent problems. The thoroughness of the exposition, the clarity of the mathematical techniques and the variety of the problems and theoretical results that are presented and rigorously analyzed make this book a primary reference in the advanced numerical analysis of partial differential equations.' Mathematical Reviews

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations Geometry Topology

Introduction	p. 1
From local to global approximation	p. 5
Comparisons of finite difference schemes	p. 9
The Fourier spectral method: first glance	p. 16
Trigonometric polynomial approximation	p. 19
Trigonometric polynomial expansions	p. 19
Discrete trigonometric polynomials	p. 24
Approximation theory for smooth functions	p. 34
Fourier spectral methods	p. 43
Fourier-Galerkin methods	p. 43
Fourier-collocation methods	p. 48
Stability of the Fourier-Galerkin method	p. 52
Stability of the Fourier-collocation method for hyperbolic problems I	p. 54
Stability of the Fourier-collocation method for hyperbolic problems II	p. 58
Stability for parabolic equations	p. 62
Stability for nonlinear equations	p. 64
Further reading	p. 65
Orthogonal polynomials	p. 66
The general Sturm-Liouville problem	p. 67
Jacobi polynomials	p. 69
Polynomial expansions	p. 79
The continuous expansion	p. 79
Gauss quadrature for ultraspherical polynomials	p. 83
Discrete inner products and norms	p. 88
The discrete expansion	p. 89
Polynomial approximation theory for smooth functions	p. 109
The continuous expansion	p. 109
The discrete expansion	p. 114
Polynomial spectral methods	p. 117
Galerkin methods	p. 117
Tau methods	p. 123
Collocation methods	p. 129
Penalty method boundary conditions	p. 133
Stability of polynomial spectral methods	p. 135
The Galerkin approach	p. 135
The collocation approach	p. 142
Stability of penalty methods	p. 145
Stability theory for nonlinear equations	p. 150
Further reading	p. 152
Spectral methods for nonsmooth problems	p. 153
The Gibbs phenomenon	p. 154
Filters	p. 160
The resolution of the Gibbs phenomenon	p. 174
Linear equations with discontinuous solutions	p. 182
Further reading	p. 186
Discrete stability and time integration	p. 187
Stability of linear operators	p. 188
Standard time integration schemes	p. 192
Strong stability preserving methods	p. 197
Further reading	p. 202
Computational aspects	p. 204
Fast computation of interpolation and differentiation	p. 204
Computation of Gaussian quadrature points and weights	p. 210
Finite precision effects	p. 214
On the use of mappings	p. 225
Spectral methods on general grids	p. 235
Representing solutions and operators on general grids	p. 236
Penalty methods	p. 238
Discontinuous Galerkin methods	p. 246
References and further reading	p. 248
Elements of convergence theory	p. 249
A zoo of polynomials	p. 252
Legendre polynomials	p. 252
Chebyshev polynomials	p. 255
Bibliography	p. 260
Index	p. 272
Table of Contents provided by Ingram. All Rights Reserved.

Spectral Methods for Time-Dependent Problems

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Hardcover

Industry Reviews

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