Description
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.
Chapter
2 Trigonometric polynomial approximation
2.1 Trigonometric polynomial expansions
2.1.1 Differentiation of the continuous expansion
2.2 Discrete trigonometric polynomials
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
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
4.1 The general Sturm–Liouville problem
4.2.1 Legendre polynomials
4.2.2 Chebyshev polynomials
4.2.3 Ultraspherical polynomials
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
6 Polynomial approximation theory for smooth functions
6.1 The continuous expansion
6.2 The discrete expansion
7 Polynomial spectral 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
9 Spectral methods for nonsmooth problems
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
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
10.3.2 SSP methods for linear operators
10.3.3 Optimal SSP Runge–Kutta methods for nonlinear problems
10.3.4 SSP multi-step methods
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
12 Spectral methods on general grids
12.1 Representing solutions and operators on general grids
12.2.2 Collocation methods
12.2.3 Generalizations of penalty methods
12.3 Discontinuous Galerkin methods
Appendix A Elements of convergence theory
Appendix B A zoo of polynomials
B.1.1 The Legendre expansion
B.1.2 Recurrence and other relations
B.2 Chebyshev polynomials
B.2.1 The Chebyshev expansion
B.2.2 Recurrence and other relations
Spectral Methods for Time-Dependent Problems
( Cambridge Monographs on Applied and Computational Mathematics )
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