Symbolic Computation of High Order Compact Difference
Schemes for Three Dimensional Linear Elliptic Partial
Differential Equations with Variable Coefficients

Jun Zhang
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA

Lexin Ge
Center for Computational Sciences
University of Kentucky
Lexington, KY 40506-0045, USA


We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.

Key words: elliptic partial differential equations, Maple software package, fourth order compact difference scheme, symbolic derivation method.

Mathematics Subject Classification: 65M06, 65N12.

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This paper has been published in Journal of Computational and Applied Mathematics, Vol. 143, No. 1, pp. 9-27 (2002).

Technical Report 299-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. This research was supported in part by the U.S. National Science Foundation under the grant CCR-9902022, and in part by the University of Kentucky Center for Computational Sciences.