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The computational efficiency and taylor expansion derivation of approximating equations to the groundwater flow equation
Author(s) -
Kuiper Logan K.
Publication year - 1978
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr014i006p01171
Subject(s) - mathematics , differential equation , flow (mathematics) , independent equation , variable (mathematics) , mathematical analysis , ordinary differential equation , space (punctuation) , groundwater flow equation , taylor series , partial differential equation , groundwater flow , geometry , groundwater , aquifer , geotechnical engineering , linguistics , philosophy , engineering
A general procedure is presented for the derivation of the approximating equations to a differential equation. As is the case with many such approximating equations, values of the dependent variable evaluated at a cluster of points in the independent variable space are related in a relatively simple manner. The total set of such relations, each with a corresponding cluster, one for each of a set of chosen points in the independent variable space, forms the approximating equations to the differential equation. In the procedure presented, each of these relations is formulated by the use of Taylor expansion about a point centrally located within each corresponding cluster. The procedure is used to derive new approximating equations to the one‐dimensional groundwater flow equation, both homogeneous and nonhomogeneous. These approximating equations are shown to exhibit superior accuracy and speed in comparison with others that are in common use presently. Twelve different varieties of approximating equations to the one‐dimensional homogeneous groundwater flow equation, many of which are in common use, are compared for accuracy. Results of these comparisons and others for the nonhomogeneous case show that approximating equations with higher‐order error e give superior results. For any given variety of approximating equations it is shown that maximum accuracy is achieved when the quantity g , the magnitude of the spacial increment Δ x squared divided by the magnitude of the time increment Δ t , is properly chosen. What this optimal value of g should be is shown to depend upon the error e and character of the approximating equations, the variability and accuracy of the solution being obtained, and the variability of the transmissivity and storage coefficient.