Summary
We derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains. In particular, we consider Galerkin and collocation methods based on Hermite functions as well as rational functions (a Fourier series combined with a cotangent mapping). We show that (i) first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues, (ii) for the Hermite method the eigenvalues are determined by the roots of the Hermite polynomials and for the rational method they are determined by the Laguerre polynomials, and (iii) the Hermite method has attractive stability properties in the sense of small condition numbers and spectral radii.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (1972): Handbook of Mathematical Functions. Dover, New York
Akhiezer, N.I. (1956): Theory of Approximation. Ungar, New York
Boyd, J. (1982): The optimization of convergence for Chebyshev polynomial methods in an unbounded domain. J. Comput. Phys.45, 43–79
Boyd, J. (1984): The asymptotic coefficients of Hermite function series. J. Comput. Phys.54, 382–410
Boyd, J. (1987): Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys.69, 112–142
Boyd, J. (1988): Chebyshev domain truncation is inferior to Fourier domain truncation for solving problems on an infinite interval. J. Sci. Comput.3, 109–120
Boyd, J. (1989): Chebyshev & Fourier Spectral Methods. Springer, Berlin Heidelberg New York
Boyd, J. (1990): The orthogonal rational functions of Higgins and Christov and algebraically mapped Chebyshev polynomials. J. Approx. Theory61, 98–105
Christov, C.I. (1982): A complete orthonormal system of functions inL 2(−∞, ∞) space. SIAM J. Appl. Math.42, 1337–1344
Christov, C.I., Bekyarov, K.L. (1990): A Fourier-series, method for solving soliton problems. SIAM J. Sci. Stat. Comput.11, 631–647
Cain, A.B., Ferziger, J.H., Reynolds, W.C. (1984): Discrete orthogonal function expansions for non-uniform grids using the fast Fourier transform. J. Comput. Phys.56, 272–286
Higgins, J.R. (1977): Completeness and basis functions of sets of special functions. Cambridge University Press, Cambridge
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A. (1987): Spectral Methods in Fluid Dynamics. Springer, Berlin Heidelberg New York
Funaro, D., Kavian, O. (1991): Approximations of some diffusion evolution equations in unbounded domains by Hermite functions. Math. Comput.57, 597–619
Gottlieb, D., Hussaini, M.Y., Orszag, S.A. (1984): Theory, and applications of spectral methods. In: R.G. Voigt (ed.) Spectral Methods for Partial Differential Equations, pp. 1–54. SIAM, Philadelphia
Grenander, U., Szegö, G. (1958): Toeplitz Forms and their Applications. University of California Press, Berkeley
Grosch, C.E., Orszag, S.A. (1977): Numerical solution of problems in unbounded regions: coordinate transformations. J. Comput. Phys.25, 274–296
Lubinsky, D.S.: Private communication
Reddy, S., Trefethen, L.N. (1990): Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues. In: C. Canuto, A. Quarteroni, eds., Spectral and High Order Methods for Partial Differential Equations, pp. 147–164. North-Holland, Amsterdam
Stenger, F., (1981): Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Review23, 165–224
Stenger, F. (1979): A “Sinc-Galerkin” method of solution of boundary value problems. Math. Comput.33, 85–109
Stenger, F.: Private communication
Szegö, G. (1939): Orthogonal Polynomials. American Mathematical Society, Providence
Trefethen, L.N.: Non-normal matrices and pseudo-eigenvalues. Preprint
Trefethen, L.N. (1992): Pseudospectra of matrices. In: D.F. Griffiths, G.A. Watson, eds., Proceedings of the 14th Dundee Biennial Conference on Numerical Analysis
Vandeven, H. (1990): On the eigenvalues of second-order spectral differentiation matrices. In: C. Canuto, A. Quarteroni, eds., Spectral and High Order Methods for Partial Differential Equations, pp. 313–318. North-Holland, Amsterdam
Weideman, J.A.C., Cloot, A. (1990): Spectral methods and mappings for evolution equations on the infinite line. In: C. Canuto, A. Quarteroni, eds., Spectral and High Order Methods for Partial Differential Equations, pp. 467–481. North-Holland, Amsterdam
Weideman, J.A.C., Trefethen, L.N. (1988): The eigenvalues of second-order spectral differentiation matrices. SIAM J. Numer. Anal.25, 1279–1298
Wilkinson, J.H. (1965): The Algebraic Eigenvalue Problem. Claredon Press, Oxford
Yuen, H.C., Ferguson, W.E. (1978): Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids.21, 1275–1278