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Spectral methods for time-dependent problems. (English) Zbl 1111.65093
Cambridge Monographs on Applied and Computational Mathematics 21. Cambridge: Cambridge University Press (ISBN 978-0-521-79211-0/hbk). ix, 273 p. (2007).
Spectral methods are well suited to solve problems modeled by time-dependent partial differential equations. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as through more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including thorough 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 of the later chapters are devoted to material not previously covered in book form. In particular stability theory for polynomial methods, techniques for dealing with problems with discontinuous solutions, round-off errors, and the formulation of spectral methods on general grids will be especially helpful for practioners.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Kxx Parabolic equations and parabolic systems
35Lxx Hyperbolic equations and hyperbolic systems
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