Abstract
A domain decomposition method is examined to solve a time-dependent parabolic equation. The method employs an orthogonal polynomial collocation technique on multiple subdomains. The subdomain interfaces are approximated with the aid of a penalty method. The time discretization is implemented in an explicit/implicit finite difference method. The subdomain interface is approximated using an explicit Dufort-Frankel method, while the interior of each subdomain is approximated using an implicit backwards Euler's method. The principal advantage to the method is the direct implementation on a distributed computing system with a minimum of interprocessor communication. Theoretical results are given for Legendre polynomials, while computational results are given for Chebyshev polynomials. Results are given for both a single processor computer and a distributed computing system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1988).Spectral Methods in Fluid Dynamics, Springer-Verlag, New York.
Davis, P. J., and Rabinowitz, P. (1984).Methods of Numerical Integration, Academic Press (Harcourt Brace Jovanovich), New York, second edition.
Funaro, D. (1985). Domains Decomposition Methods for Pseudo-Spectral Approximations, Technical report, Univeritá degli studi di Pavia, Strada Nuova 65, 27100 Pavia, Italy; research supported in part by AFOSR grant No. 85-0303.
Funaro, D. (1986). A multidomain spectral approximation of elliptic equations,Numer. Methods P.D.E. 2, 187–205.
Gottlieb, D., and Gustafsson, B. (1975). Generalized Dufort-Frankel Methods for Parabolic Initial-Boundary-Value Problems, ICASE Report No. 75-5, ICASE, NASA Langley Research Center.
Gottlieb, D., and Lustman, L. (1981). The Dufort-Frankel Chebyshev Method for Parabolic Initial Boundary Value Problems, ICASE Report No. 81-42, ICASE, NASA Langley Research Center.
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, Society of Industrial and Applied Mathematics, Philadelphia, Pennsylvania.
Gottlieb, D., and Hirsh, R. S. (1989). Parallel pseudospectral domain decomposition techniques,J. Sci. Comput. 4(4), 309–325.
Gottlieb, D., and Lustman, L. (1983). The spectrum of the Chebyshev collocation operator for the heat equation,SIAM J. Numer. Anal. 20(5), 909–921.
Marchuk, G. I. (1975).Methods of Numerical Mathematics, Springer-Verlag, New York.
Quarteroni, A. (1989). Domain Decomposition Methods for Systems of Conservation Laws: Spectral Collocation Approximations, ICASE Report No. 89-5, ICASE, NASA Langley Research Center.
Vandeven, H. (1989). On the eigenvalues of second-order spectral, differential operators. In Canuto, C., and Quarteroni, A. (eds.),Spectral and High-Order Methods for Partial Differential Equations, pp. 313–318, North-Holland, Amsterdam; Proceedings of the ICOSAHOM 89 Conference.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Black, K. Polynomial collocation using a domain decomposition solution to parabolic PDE's via the penalty method and explicit/implicit time marching. J Sci Comput 7, 313–338 (1992). https://doi.org/10.1007/BF01108035
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01108035