Abstract
We study Chebyshev collocation when applied to a system of symmetric hyperbolic equations on a finite domain with general boundary conditions. We show that the use of orthogonal projections in theL 2 norm in order to smooth out the higher modes and to implement boundary conditions leads to a stable numerical approximation in theL 2 norm; the stability estimate corresponds to the estimate of the continuous problem. For constant coefficient systems the method reduces to an efficient implementation of Legendre-Galerkin.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Canuto, C., and Quarteroni, A. (1982). Approximation results for orthogonal polynomials in Sobolev spaces,Math. Comp. 38, 67–86.
Gottlieb, D. (1981). The stability of pseudospectral-Chebyshev methods,Math. Comp. 36, 107–118.
Gottlieb, D., Lustman, L., and Tadmor, E. (1986). Stability analysis of spectra for hyperbolic initial-boundary value problems,SIAM J. Numer. Anal. 23, 1–10.
Gottlieb, D., Lustman, L., and Tadmor, E. (1986). Convergence of Spectral Methods for Hyperbolic Initial-Boundary Value Systems, ICASE Report, January.
Gottlieb, D., Lustman, L., and Turkel, E. (1981). Stability of pseudospectral and finitedifference methods for variable coefficients problems,Math. Comp. 37, 293–305.
Gottlieb, D., and Orszag, S. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, Regional Conf. Series in Appl. Math., SIAM, Philadelphia, Pennsylvania.
Kreiss, H.-O., and Oliger, J. (1979). Stability of the Fourier method,Math. Comp. 16, 421–433.
Friedrichs, K. C. (1954). Symmetric hyperbolic linear differential equations,Commun. Pure Appl. Math. 7, 345–392.
Lax, P., and Richtmyer, R. (1956). Survey of the stability of linear finite difference equations,Commun. Pure Appl. Math. 9, 267.
Majda, A., McDonough, J., and Osher, S. (1978). The Fourier method for nonsmooth initial data,Math. Comp. 32, 1041–1081.
Quarteroni, A. (1982).Theoretical and Computational Aspects of Spectral Methods, Proc. Fifth International Conf. of Computing Methods in Appl. Sciences and Engineering, North-Holland, Amsterdam.
Tadmor, E. (1986). The Exponential Accuracy of Fourier and Chebyshev Differencing Methods, ICASE Report No. 86-2, January.
Thomée, V. (1957). Estimates of the Friedrichs-Lewy type for mixed problems in the theory of linear hyperbolic differential equations in two independent variables,Math. Scand. 5, 93–113.
Whittaker, E., and Watson, G. (1927).A Course in Modern Analysis, Cambridge Univ. Press, Cambridge.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Reyna, L.G. L 2 estimates for Chebyshev collocation. J Sci Comput 3, 1–23 (1988). https://doi.org/10.1007/BF01066480
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01066480