Book contents
- Frontmatter
- Contents
- Preface
- Abbreviations
- 1 Introduction
- 2 Introduction to spectral methods via orthogonal functions
- 3 Introduction to PS methods via finite differences
- 4 Key properties of PS approximations
- 5 PS variations and enhancements
- 6 PS methods in polar and spherical geometries
- 7 Comparisons of computational cost for FD and PS methods
- 8 Applications for spectral methods
- Appendices
- References
- Index
3 - Introduction to PS methods via finite differences
Published online by Cambridge University Press: 03 December 2009
- Frontmatter
- Contents
- Preface
- Abbreviations
- 1 Introduction
- 2 Introduction to spectral methods via orthogonal functions
- 3 Introduction to PS methods via finite differences
- 4 Key properties of PS approximations
- 5 PS variations and enhancements
- 6 PS methods in polar and spherical geometries
- 7 Comparisons of computational cost for FD and PS methods
- 8 Applications for spectral methods
- Appendices
- References
- Index
Summary
Both periodic and nonperiodic PS methods can be seen as high-accuracy limits of FD methods. This alternative approach to PS methods provides both generalizations and insights.
Orthogonal polynomials and functions lead only to a small class of possible spectral methods, whereas the FD viewpoint allows many generalizations.
For example, all classical orthogonal polynomials cluster the nodes quadratically toward the ends of the interval – this is often, but not always, best.
An FD viewpoint offers a chance to explore intermediate methods between low-order FD and PS methods.
One might consider procedures of not quite as high order as Chebyshev spectral methods, and with nodes not clustered quite as densely – possibly trading some accuracy for stability and simplicity.
Two separate ways to view any method will always provide more opportunities for analysis, understanding, and improvement.
Many special enhancements have been designed for FD methods. Viewing PS methods as a special case of FD methods often makes it easier to carry over such ideas.
Examples include staggered grids, upwind techniques, boundary techniques, polar and spherical coordinates, etc. (see Chapters 5 and 6).
Comparisons between PS and FD methods can be made more consistent.
Sections 3.1 and 3.2 contain some general material on FD approximations, allowing us in Section 3.3 to discuss different types of node distributions. The relation between these types and the accuracy that polynomial interpolation provides (at different locations over [–1, 1]) is clarified in Section 3.4.
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- A Practical Guide to Pseudospectral Methods , pp. 14 - 35Publisher: Cambridge University PressPrint publication year: 1996