Abstract
Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula cannot be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula; however, we show in this article that this is the wrong choice and it may lead to divergence if time-dependent methods are used to march the solution to steady state. We develop, in this paper, the correct quadrature formula for these problems. This formula takes into account the degree of the polynomials involved. We show that this formula leads to a well-conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.
Similar content being viewed by others
References
Canuto, G., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1987).Spectral Methods in Fluid Dynamics, Springer-Verlag, New York.
Davis, P. J., and Rabinowitz, P. (1984).Methods of Numerical Integration, 2nd ed., Academic Press, New York.
Gottlieb, D., Hussaini, M. Y., and Orszag, S. A. (1982). Theory and applications of spectral methods, inSpectral Methods for Partial Differential Equations, Voigt, R. G., Gottlieb, D., and Hussaini, M. Y. (eds.), SIAM-CBMS, pp. 1–54.
Orszag, S. A., Israeli, M., and Deville, M. O. (1988). Boundary conditions for incompressible flows,J. Sci. Comput. 1, 75–111.
Streett, C. L., and Macaraeg, M. G. (1989). Spectral multidomain for large-scale fluid dynamic simulations,Appl. Num. Math. 6, 123–139.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gottlieb, D., Streett, C.L. Quadrature imposition of compatibility conditions in Chebyshev methods. J Sci Comput 5, 223–239 (1990). https://doi.org/10.1007/BF01089165
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01089165