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Fluid MechanicsA Geometrical Point of View$
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S. G. Rajeev

Print publication date: 2018

Print ISBN-13: 9780198805021

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198805021.001.0001

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Spectral Methods

Spectral Methods

Chapter:
(p.175) 13 Spectral Methods
Source:
Fluid Mechanics
Author(s):

S. G. Rajeev

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198805021.003.0013

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

Keywords:   Spectral methods, Chebychev basis, Orszag, Orr–Sommerfeld equation, Trefethen, interpolation, down sampling

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