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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Hesthaven, Jan
Otros Autores: Gottlieb, Sigal, Gottlieb, David
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, ©2007.
Colección:Cambridge monographs on applied and computational mathematics ; 21.
Temas:
Spectral theory (Mathematics)
Differential equations, Partial.
Differential equations, Hyperbolic.
Spectre (Mathématiques)
Équations aux dérivées partielles.
Équations différentielles hyperboliques.
MATHEMATICS > Differential Equations > Partial.
Differential equations, Hyperbolic
Differential equations, Partial
Spektralmethode
Zeitabhängigkeit
Partielle Differentialgleichung
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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