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Spectral methods for time-dependent problems /
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 sel...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
©2007.
|
| Colección: | Cambridge monographs on applied and computational mathematics ;
21. |
| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 100 | 1 | |a Hesthaven, Jan. | |
| 245 | 1 | 0 | |a Spectral methods for time-dependent problems / |c Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb. |
| 260 | |a Cambridge : |b Cambridge University Press, |c ©2007. | ||
| 300 | |a 1 online resource (ix, 273 pages) : |b illustrations | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 490 | 1 | |a Cambridge monographs on applied and computational mathematics ; |v 21 | |
| 504 | |a Includes bibliographical references and index. | ||
| 588 | 0 | |a Print version record. | |
| 520 | |a 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. | ||
| 505 | 0 | |a Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Introduction -- 1 From local to global approximation -- 1.1 Comparisons of finite difference schemes -- 1.1.1 Phase error analysis -- 1.1.2 Finite-order finite difference schemes -- 1.1.3 Infinite-order finite difference schemes -- 1.2 The Fourier spectral method: first glance -- 1.3 Further reading -- 2 Trigonometric polynomial approximation -- 2.1 Trigonometric polynomial expansions -- 2.1.1 Differentiation of the continuous expansion -- 2.2 Discrete trigonometric polynomials -- 2.2.1 The even expansion -- 2.2.2 The odd expansion -- 2.2.3 A first look at the aliasing error -- 2.2.4 Differentiation of the discrete expansions -- 2.3 Approximation theory for smooth functions -- 2.3.1 Results for the continuous expansion -- 2.3.2 Results for the discrete expansion -- 2.4 Further reading -- 3 Fourier spectral methods -- 3.1 Fourier-Galerkin methods -- 3.2 Fourier-collocation methods -- 3.3 Stability of the Fourier-Galerkin method -- 3.4 Stability of the Fourier-collocation method for hyperbolic problems I -- 3.5 Stability of the Fourier-collocation method for hyperbolic problems II -- 3.6 Stability for parabolic equations -- 3.7 Stability for nonlinear equations -- 3.8 Further reading -- 4 Orthogonal polynomials -- 4.1 The general Sturm-Liouville problem -- 4.2 Jacobi polynomials -- 4.2.1 Legendre polynomials -- 4.2.2 Chebyshev polynomials -- 4.2.3 Ultraspherical polynomials -- 4.3 Further reading -- 5 Polynomial expansions -- 5.1 The continuous expansion -- 5.1.1 The continuous legendre expansion -- 5.1.2 The continuous Chebyshev expansion -- 5.2 Gauss quadrature for ultraspherical polynomials -- 5.2.1 Quadrature for Legendre polynomials -- 5.2.2 Quadrature for Chebyshev polynomials -- 5.3 Discrete inner products and norms -- 5.4 The discrete expansion. | |
| 505 | 8 | |a 5.4.1 The discrete Legendre expansion -- 5.4.2 The discrete Chebyshev expansion -- 5.4.3 On Lagrange interpolation, electrostatics, and the Lebesgue constant -- 5.5 Further reading -- 6 Polynomial approximation theory for smooth functions -- 6.1 The continuous expansion -- 6.2 The discrete expansion -- 6.3 Further reading -- 7 Polynomial spectral methods -- 7.1 Galerkin methods -- 7.2 Tau methods -- 7.3 Collocation methods -- 7.4 Penalty method boundary conditions -- 8 Stability of polynomial spectral methods -- 8.1 The Galerkin approach -- 8.2 The collocation approach -- 8.3 Stability of penalty methods -- 8.4 Stability theory for nonlinear equations -- 8.5 Further reading -- 9 Spectral methods for nonsmooth problems -- 9.1 The Gibbs phenomenon -- 9.2 Filters -- 9.2.1 A first look at filters and their use -- 9.2.2 Filtering Fourier spectral methods -- 9.2.3 The use of filters in polynomial methods -- 9.2.4 Approximation theory for filters -- 9.3 The resolution of the Gibbs phenomenon -- 9.4 Linear equations with discontinuous solutions -- 9.5 Further reading -- 10 Discrete stability and time integration -- 10.1 Stability of linear operators -- 10.1.1 Eigenvalue analysis -- 10.1.2 Fully discrete analysis -- 10.2 Standard time integration schemes -- 10.2.1 Multi-step schemes -- 10.2.2 Runge-Kutta schemes -- 10.3 Strong stability preserving methods -- 10.3.1 SSP theory -- 10.3.2 SSP methods for linear operators -- 10.3.3 Optimal SSP Runge-Kutta methods for nonlinear problems -- 10.3.4 SSP multi-step methods -- 10.4 Further reading -- 11 Computational aspects -- 11.1 Fast computation of interpolation and differentiation -- 11.1.1 Fast Fourier transforms -- 11.1.2 The even-odd decomposition -- 11.2 Computation of Gaussian quadrature points and weights -- 11.3 Finite precision effects -- 11.3.1 Finite precision effects in Fourier methods. | |
| 505 | 8 | |a 11.3.2 Finite precision in polynomial methods -- 11.4 On the use of mappings -- 11.4.1 Local refinement using Fourier methods -- 11.4.2 Mapping functions for polynomial methods -- 11.5 Further reading -- 12 Spectral methods on general grids -- 12.1 Representing solutions and operators on general grids -- 12.2 Penalty methods -- 12.2.1 Galerkin methods -- 12.2.2 Collocation methods -- 12.2.3 Generalizations of penalty methods -- 12.3 Discontinuous Galerkin methods -- 12.4 Further reading -- Appendix A Elements of convergence theory -- Appendix B A zoo of polynomials -- B.1 Legendre polynomials -- B.1.1 The Legendre expansion -- B.1.2 Recurrence and other relations -- B.1.3 Special values -- B.1.4 Operators -- B.2 Chebyshev polynomials -- B.2.1 The Chebyshev expansion -- B.2.2 Recurrence and other relations -- B.2.3 Special values -- B.2.4 Operators -- Bibliography -- Index. | |
| 590 | |a eBooks on EBSCOhost |b EBSCO eBook Subscription Academic Collection - Worldwide | ||
| 650 | 0 | |a Spectral theory (Mathematics) | |
| 650 | 0 | |a Differential equations, Partial. | |
| 650 | 0 | |a Differential equations, Hyperbolic. | |
| 650 | 6 | |a Spectre (Mathématiques) | |
| 650 | 6 | |a Équations aux dérivées partielles. | |
| 650 | 6 | |a Équations différentielles hyperboliques. | |
| 650 | 7 | |a MATHEMATICS |x Differential Equations |x Partial. |2 bisacsh | |
| 650 | 7 | |a Differential equations, Hyperbolic |2 fast | |
| 650 | 7 | |a Differential equations, Partial |2 fast | |
| 650 | 7 | |a Spectral theory (Mathematics) |2 fast | |
| 650 | 7 | |a Spektralmethode |2 gnd | |
| 650 | 7 | |a Zeitabhängigkeit |2 gnd | |
| 650 | 7 | |a Partielle Differentialgleichung |2 gnd | |
| 700 | 1 | |a Gottlieb, Sigal. | |
| 700 | 1 | |a Gottlieb, David. | |
| 776 | 0 | 8 | |i Print version: |a Hesthaven, Jan. |t Spectral methods for time-dependent problems. |d Cambridge : Cambridge University Press, ©2007 |z 9780521792110 |z 0521792118 |w (DLC) 2007276026 |w (OCoLC)70764827 |
| 830 | 0 | |a Cambridge monographs on applied and computational mathematics ; |v 21. | |
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