Backward difference operator

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

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 relationship between backward difference operators and differential operators can now be established. Combine Eqs. (3.22) and (3.24) to obtain... 

The higher-order backward difference operator, V", can be obtained by raising... 

Expansion of the exponential terms and rearrangement yields the following equations for the second and third backward difference operators ... 

Equations (3.32), (3.36), and (3.37) express the backward difference operators in terms of infinite series of differential operators. In order to complete the set of relationships, equations that express the differential operators in terms of backward difference operators will also be derived. To do so, first rearrange Eq. (3.31) to solve for e ... 

The relationships between backward difference operators and differential operators, which are summarized in Table 3.1, enable us to develop a variety of formulas expressing derivatives of functions in terms of backward finite differences, and vice versa. In addition, these formulas may have any degree of accuracy desired, provided that a sufficient number of terms is retained in the manipulation of these infinite series. This concept will be demonstrated in the remainder of this section. 

In this set of equations a prime has been used to denote the derivative to simplify the notation. Also no assumptions have been made with regard to the form of the equations. In each equation it is assumed that all the function values and derivatives may possible be present. Also it is assumed that any of the terms may appear in any nonlinear or implicit form. It is only assumed that the equations are written in a form that gives zero as the answer when satisfied by the set of variables and derivatives and at each time point. To update this set of coupled differential equations at some time point using the known value of the function at some previous time point simply requires that the derivative terms be replaced by one of the previously discussed three approximations. For example, if the backwards difference operator of Eq. (10.4) is used the replacement is ... 

To facilitate the development of explicit and implicit methods, it is necessary to briefly consider the origins of interpolation and quadrature formulas (i.e., numerical approximation to integration). There are essentially two methods for performing the differencing operation (as a means to approximate differentiation) one is the forward difference, and the other is the backward difference. Only the backward difference is of use in the development of eiqjlicit and implicit methods. 

Consideration was given to whether the model would be better run on a forward-, backward-or central-difference principle, however the convenient use of elemental length values considerably shorter than the maximum for consistent operation effectively eliminated any significant difference. A backward-difference approach was used. 

With these introductory concepts in mind, let us proceed to develop the backward, forward, and central difference operators and the relationships between these and the differential operators. 

These relationships can also be obtained by combining the definitions of the backward differences [Eqs, and (3.26)1 with the definition of the inverse shift operator [Eqs, (3.11) and (3.23)1,... 

The complete set of relationships between central difference operators and differential operators is summarized in Table 3.3. These relationships will be used in Chap. 4 to develop a set of formulas expressing the derivatives in terms of central finite differences. These formulas will have higher accuracy than those developed using backward and forward finite differences. 

In hyperspherical coordinates, the wave function changes sign when <]) is increased by 2k. Thus, the cotTect phase beatment of the (]) coordinate can be obtained using a special technique [44 8] when the kinetic energy operators are evaluated The wave function/((])) is multiplied with exp(—i(j)/2), and after the forward EFT [69] the coefficients are multiplied with slightly different frequencies. Finally, after the backward FFT, the wave function is multiplied with exp(r[Pg.60]

A backward-curved impeller blade combines all these effects. The exit velocity triangle for this impeller with the different slip phenomenon changes is shown in Figure 6-25. This triangle shows that actual operating conditions are far removed from the projected design condition. 

Figure 12-119E. AMCA standard operating limits classes for single width centrifugal fans—ventilating airfoils and backwardly inclined. Other designs with different strength parameters are defined in the same publication noted. (Reprinted by permission A/WCA Publication 99-86Standards Handbook, 1986, Standard AMCA No. 99-2408-69, with written permission from Air Movement and Control Association International, Inc., 1986. All rights reserved.)...

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