Central difference operator averaged

D = differential operator I = integral operator E = shift operator A = forward difference operator V = backward difference operator 6 = central difference operator p = averager operator. 

The relationships between central difference operators and differential operators can now be developed, Eq. (3.73), representing the first averaged central difference, is combined with Eqs. (3.17) and (3.22) to yield... 

The higher-order averaged odd central difference operators are obtained by taking products of Eqs. (3.78) and (3.81). The higher-order even central differences are formulated by taking powers of Eq. (3.81). The third and fourth central operators, thus obtained, are listed below ... 

Express the differential and averaged central difference operators in terms of Iheir respective definitions ... 

This shows that the central-difference approximation can be thought of as an operator that filters out scales smaller than the mesh size. Furthermore, the approximation yields the derivative of an averaged value of v x). 

The odd-order differential operators in Eq. (3.127) are replaced by averaged central differences and the even-order differential operators by central differences, all taken from Table 3.3. Substituting these into Eq. (3.127) and regrouping of terms yield the formula... 

Other forms of Stirling s interpolation formula exist, which make use of base points spaced at half intervals (i.e., at h/2). Our choice of using averaged central differences to replace the odd differential operators eliminated the need for having base points located at the midpoints. The central differences for Eq. (3.129) are tabulated in Table 3.6. 

Horticulture Pit. The horticulture pit was constructed in 1969-70 and has been in continuous use since that time. The 8.8 x 3.4 m concrete pit has an average depth of 1 m and has been filled with alternate 30 cm layers of gravel and soil as shown in Figure 1. The soil is a silt loam (Clarion-Nicolett-Webster) characteristic of Central Iowa. A cover closes automatically to prevent precipitation from entering. A tile system constructed below the concrete floor of the pit connects to a sump for sampling of the ground water. The 30,000 L pit has had over 30 kg of more than 40 different pesticides deposited since beginning operation. 

Temporal differentiation, on the other hand, is approximated by central finite-difference (< /, 2ti <52) and central average (/xt, /i2t) operators defined in Table 5.1 [21]. 

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