Comparison Of A Variational Method With A Finite-Difference Method For The Problem Of Natural Convection In Porous Media
- P.H. Holst (Department of Chemical Engineering) | K. Aziz (Department of Chemical Engineering)
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- Society of Petroleum Engineers
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- Document Type
- 1969. Society of Petroleum Engineers
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Unsolicited. This document was submitted to SPE (or its predecessor organization) for consideration for publication in one of its technical journals. While not published, this paper has been included in the eLibrary with the permission of and transfer of copyright from the author.
Numerical methods as applied to the solution of complex mathematical equations have been becoming increasingly sophisticated. Some of the latest developments utilize variational methods in the solution of partial differential equations, Several investigators have found that for some problems the variational techniques yield considerable improvements in problems the variational techniques yield considerable improvements in accuracy over standard finite difference methods. In this investigation similar variational techniques are applied to the solution of a system of two nonlinear partial differential equations. The equations, one elliptic equation describing flow and one parabolic equation describing temperature are obtained by combining Darcy's law with the equations of energy and mass conservation. Two dimensional solutions are obtained by standard finite difference techniques (ADIP), and variational methods using piece-wise linear basis functions. The solutions are obtained for two sets of boundary conditions (vertically heated and horizontally heated) and subsequently compared. From the comparison of the two methods it is concluded that the finite difference method is superior to the variational method for this problem. The amount of preparatory work involved in obtaining a variational solution was found to be considerably larger than for the finite difference method even for the simple basis functions considered here.
With the advent of high speed digital computers, a means of numerically solving complex mathematical models became a reality. Initially most numerical solutions were based on some finite difference scheme such as successive over-relaxation (SOR), alternating direction implicit (ADIP), alternating direction explicit (ADEP), etc.
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