Coverart for item
The Resource Spectral methods for time-dependent problems, Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb

Spectral methods for time-dependent problems, Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb

Label
Spectral methods for time-dependent problems
Title
Spectral methods for time-dependent problems
Statement of responsibility
Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb
Creator
Contributor
Subject
Language
eng
Summary
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
Member of
Cataloging source
N$T
http://library.link/vocab/creatorName
Hesthaven, Jan
Dewey number
515.3535
Illustrations
illustrations
Index
index present
LC call number
QC20.7.S64
LC item number
H47 2007eb
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/relatedWorkOrContributorName
  • Gottlieb, Sigal
  • Gottlieb, David
Series statement
Cambridge monographs on applied and computational mathematics
Series volume
21
http://library.link/vocab/subjectName
  • Spectral theory (Mathematics)
  • Differential equations, Partial
  • Differential equations, Hyperbolic
  • MATHEMATICS
  • Differential equations, Hyperbolic
  • Differential equations, Partial
  • Spectral theory (Mathematics)
  • Spektralmethode
  • Zeitabhängigkeit
  • Partielle Differentialgleichung
Label
Spectral methods for time-dependent problems, Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb
Instantiates
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • 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
  • 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
  • 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
Control code
162144521
Dimensions
unknown
Extent
1 online resource (ix, 273 pages)
File format
unknown
Form of item
online
Isbn
9780511260506
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other physical details
illustrations
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
(OCoLC)162144521
Label
Spectral methods for time-dependent problems, Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • 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
  • 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
  • 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
Control code
162144521
Dimensions
unknown
Extent
1 online resource (ix, 273 pages)
File format
unknown
Form of item
online
Isbn
9780511260506
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other physical details
illustrations
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
(OCoLC)162144521

Library Locations

    • Thomas Jefferson LibraryBorrow it
      1 University Blvd, St. Louis, MO, 63121, US
      38.710138 -90.311107
Processing Feedback ...