Abstract
It is well known that spectral approximations of hyperbolic time-dependent equations can lead to incoherent results in the case where the solution is discontinuous. However, it has been proved that the spectral coefficients of the approximation are computed precisely. In this article we present and analyze a class of filters that allows the recovering of the solution with an exponential accuracy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abarbanel, S., Gottlieb, D., and Tadmor, E. (1985). Spectral Methods for Discontinuous Problems, ICASE Report N° 85-38, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Abramowitz, M., and Stegun, I. A. (eds.) (1970).Handbook of Mathematical Functions, Dover, New York.
Bass, J. (1956).Cours de Mathématiques, Masson.
Canuto, C., and Quarteroni, A. (1982). Error estimates for spectral and pseudospectral approximations of hyperbolic equations,SIAM J. Numer. Anal. 19(3), 629–642.
Davis, P. J., and Rabinowitz, P. (1975).Methods of Numerical Integration, Academic Press New York, San Francisco, and London.
Fornberg, B. (1985). On a Fourier method for the integration of hyperbolic equations,SIAM J. Numer. Anal. 12(3), 509–528.
Gottlieb, D., Lustman, L., and Tadmor, E. (1986). Stability Analysis of Spectral Methods for Hyperbolic Initial-Boundary Value System, ICASE Report N° 86-2, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, CBMS Regional Conference series in applied Mathematics26,SIAM, Philadelphia.
Gottlieb, D., and Tadmor, E. (1985). Recovering Pointwise Value of Discontinuous Data within Spectral Accuracy, ICASE Report N° 85-3, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Gottlieb, D., Orszag, S. A., and Turkel, E. (1979). Stability of Pseudospectral and Finite Difference Methods for Variable Coefficient Problems, ICASE Report N° 79-32, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Hussaini, M. Y., Kopriva, D. A., Salas, M. D., and Zang, T. A. (1984). Spectral Methods for the Euler Equations: Fourier Methods and Shock-Capturing, ICASE Report N° 84-3, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Hussaini, M. Y., Kopriva, D. A., Salas, M. D., and Zang, T. A. (1985). Spectral Methods for the Euler Equations: Fourier Methods and Shock-Capturing, ICASE Report N° 85-38, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Hussaini, M. Y., Kopriva, D. A., Salas, M. D., and Zang, T. A. (1984). Spectral Methods for the Euler Equations: Chebyshev Methods and Shock-Fitting, ICASE Report N° 84-4, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Jackson, D. (1912). On the approximation by trigonometrical sums and polynomial,Trans. Amer. Math. Soc. 13, 491–515.
Kopriva, D. (1979). A Spectral Multidomain Method for the Solution of Hyperbolic Systems, ICASE Report N° 86-28, Institute for Computer Applications in Science and Engineering, Hampton, Virginia.
Kreiss, H. O., and Oliger, J. (1972). Comparaison of accurate methods for the integration of hyperbolic equations,Tellus 26, 199–215.
Pasciak, J. (1980). Spectral and pseudospectral methods for advection equations,Math. Comput. 35, 1081–1092.
Tadmor, E. (1986). The exponential accuracy of Fourier and Chebyshev differencing methods,SIAM J. Numer. Anal. 23(1), 1–10.
Zygmund, A. (1959).Trigonometric Series, Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vandeven, H. Family of spectral filters for discontinuous problems. J Sci Comput 6, 159–192 (1991). https://doi.org/10.1007/BF01062118
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01062118