Spectral Methods Based on Nonclassical Orthogonal Polynomials

Spectral methods for solving differential equations of boundary value type have traditionally been based on classical orthogonal polynomials such as the Chebyshev, Legendre, Laguerre, and Hermite polynomials. In this numerical study we show that methods based on nonclassical orthogonal polynomials may sometimes be more accurate. Examples include the solution of a two-point boundary value problem with a steep boundary layer and two Sturm-Liouville problems.

[1]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[2]  WALTER GAUTSCHI.  Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules , 1994, TOMS.

[3]  G. Golub,et al.  How to generate unknown orthogonal polynomials out of known orthogonal polynomials , 1992 .

[4]  Bruno Welfert.  Generation of Pseudospectral Differentiation Matrices I , 1997 .

[5]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

[6]  Walter Gautschi.  Gauss-type Quadrature Rules for Rational Functions , 1993 .

[7]  J. Pryce.  Numerical Solution of Sturm-Liouville Problems , 1994 .

[8]  David Elliott,et al.  Numerical Integration IV , 1993 .

[9]  Walter Gautschi,et al.  Algorithm 793: GQRAT—Gauss quadrature for rational functions , 1999, TOMS.

[10]  F. Stenger.  Numerical Methods Based on Sinc and Analytic Functions , 1993 .

[11]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[12]  Walter Van Assche,et al.  Quadrature formulas based on rational interpolation , 1993, math/9307221.

[13]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[14]  E. Tadmor.  The exponential accuracy of Fourier and Chebyshev differencing methods , 1986 .

[15]  J. A. C. Weideman,et al.  The eigenvalues of Hermite and rational spectral differentiation matrices , 1992 .

[16]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .