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Solution of ODEs and Eigenvalue Problems with a Chebyshev Polynomial Spectral Method

Mark J. McCready
Organization: University of Notre Dame
Department: Department of Chemical Engineering
URL: http://www.nd.edu/~mjm
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This notebook demonstrates the Orszag-tau (a modification of the Galerkin) spectral method for a simple ODE and an ODE eigenvalue problem. It is intended as a first introduction to solving these problems with a spectral numerical method.

Reference: S. A. Orszag (1971) "Accurate solution of the Orr-Sommerfeld stability equation", Journal of Fluid Mechanics, 50 pp 689-703.

The coefficients of the algebraic equations are computed directly from the orthogonality properties of the Chebyshev polynomials using orthogonality.

Reference: R. Miesen and B. J. Boersma (1995) "Hydrodynamic stability of a sheared liquid film", Journal of Fluid Mechanics, 301 pp 175-202.

*Mathematics > Calculus and Analysis > Differential Equations

Tau-spectral method, numerical eigenvalue problem, spectral method
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spectral_ode_eigens.nb (764.8 KB) - Mathematica notebook