A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations

Funaro, D.; Gottlieb, D.

doi:10.1090/S0025-5718-1988-0958637-X

A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations
HTML articles powered by AMS MathViewer

by D. Funaro and D. Gottlieb PDF

Math. Comp. 51 (1988), 599-613 Request permission

Abstract:

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

References

C. Canuto and A. Quarteroni, Error estimates for spectral and pseudospectral approximations of hyperbolic equations, SIAM J. Numer. Anal. 19 (1982), no. 3, 629–642. MR 656477, DOI 10.1137/0719044
Daniele Funaro, Some results about the spectrum of the Chebyshev differencing operator, Numerical approximation of partial differential equations (Madrid, 1985) North-Holland Math. Stud., vol. 133, North-Holland, Amsterdam, 1987, pp. 271–284. MR 899798, DOI 10.1016/S0304-0208(08)71738-9
Daniele Funaro, A preconditioning matrix for the Chebyshev differencing operator, SIAM J. Numer. Anal. 24 (1987), no. 5, 1024–1031. MR 909062, DOI 10.1137/0724067
Daniele Funaro, Domain decomposition methods for pseudospectral approximations. I. Second order equations in one dimension, Numer. Math. 52 (1988), no. 3, 329–344. MR 929576, DOI 10.1007/BF01398883

The Theory of Matrices

David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR 0520152
David Gottlieb, The stability of pseudospectral-Chebyshev methods, Math. Comp. 36 (1981), no. 153, 107–118. MR 595045, DOI 10.1090/S0025-5718-1981-0595045-1
David Gottlieb and Liviu Lustman, The spectrum of the Chebyshev collocation operator for the heat equation, SIAM J. Numer. Anal. 20 (1983), no. 5, 909–921. MR 714688, DOI 10.1137/0720063
Theodore J. Rivlin, The Chebyshev polynomials, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0450850

Global Collocation Methods for Approximation and the Solution of Partial Differential Equations

J. Comput. Phys.