The Finite Difference Approximation

Regarding accuracy, the finite difference approximations for the radial derivatives converge O(Ar ). The approximation for the axial derivative converges 0(Az), but the stability criterion forces Az to decrease at least as fast as Ar. Thus, the entire computation should converge O(Ar ). The proof of convergence requires that the computations be repeated for a series of successively smaller grid sizes. 

The absorbance spectrum in Figure 54-1 is made from synthetic data, but mimics the behavior of real data in that both are represented by data points collected at discrete and (usually) uniform intervals. Therefore the calculation of a derivative from actual data is really the computation of finite differences, usually between adjacent data points. We will now remove the quotation marks from around the term, and simply call all the finite-difference approximations a derivative. As we shall see, however, often data points that are more widely spread are used. If the data points are sufficiently close together, then the approximation to the true derivative can be quite good. Nevertheless, a true derivative can never be measured when real data is involved. 

Figure 55-8 Second derivatives calculated using different spacings for the finite difference approximation to the true derivative. The underlying curve is the 20 nm bandwidth absorbance band in Figure 54-1, with data points every nm. Figure 55-8a Difference spacings = 1-5 nm Figure 55-8b Spacings = 5-40 nm Figure 55-8c Spacings = 40-90 nm. (see Color Plate 22)...

The Fukui functions generalize the concept of frontier orbitals by including the relaxation of the orbital upon the net addition or removal of one electron. Because the number of electrons of an isolated system can only change by discrete integer number, the derivative in Equation 24.37 is not properly defined. Only the finite difference approximation of Equation 24.37 allows to define these Fukui functions (noted here by capital letters) F1 (r)... 

Although we will not discuss the finite difference approximations E1 (r) further in this section, it is useful to note that these functions can be defined also as potential derivatives of a gap and a chemical potential. The gap is the usual definition of the chemical hardness [8,9],... 

It is to be noted that/(r) is normalized to unity. Due to discontinuity problem in the number of electrons [13] in atoms and molecules, the right- and left-hand side derivatives at a fixed number of electrons introduces the concepts of EF for nucleophilic and electrophilic attack, respectively. Introducing the finite difference approximation and the concept of atom condensed Fukui function (CFF) [14], the working equations are... 

When faster reactions are dealt with, it may be profitable to remove the At/Ay2 < 0.5 condition and use an implicit method such as the Crank-Nicholson method.15 17 The finite difference approximation is then applied at the value of t corresponding to the middle of the j to j + 1 interval, leading to... 

As an example, we can use DFT calculations to evaluate the frequency of the stretching mode for a gas-phase CO molecule using the finite-difference approximation described above. The result from applying Eq. (5.3) for various values of the finite-difference displacement, 8b, are listed and plotted in Table 5.1 and Fig. 5.1. For a range of displacements, say from 8b 0.005 — 0.04 A, the... 

Use DFT calculations to determine the vibrational frequency of gas-phase N2. Compare your result with experimental data. How do your results depend on the displacement used in the finite-difference approximation ... 

General Properties of the FDA. To understand the properties of the finite difference approximation in the frequency domain, we may look at the properties of its. s-planc to z-plane mapping... 

Perhaps surprisingly, it will be shown in a later section that the above recursion is exact at the sample points in spite of the apparent crudeness of the finite difference approximation. 

It is interesting to compare the digital waveguide simulation technique to the recursion produced by the finite difference approximation (FDA) applied to the wave equation. Recall from (10.10) that the time update recursion for the ideal string digitized via the... 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

Numerically, the latter derivatives can be computed using the finite difference approximation ... 

The methodology in this section concerns global properties, which can be written as first or second order derivatives of the energy with respect to the number of electrons N. In practice, these derivatives cannot be calculated analytically and their numerical calculation is performed using a finite difference approximation ("faute de mieux") very recently, a variational ansatz has been proposed [61]. For the chemical potential (or minus the electronegativity), the finite difference approximation becomes ... 

The Finite Difference Approximations and the Iterative Technique. The reaction kinetics are dependent on the rest of the equations through particle temperatures. If we have an initial guess on the coal particle temperature history during its total residence in the reactor then the relations given by Eq. (15) are decoupled from the rest and can be solved separately. Consequently, the volatilization estimates, v j(t) s, can be used to update particle temperatures with Eqs. (11) through (14). 

This procedure had converged in 4 or 5 iterations to four significant figures for all cases tried in this study. The accuracy of the calculations depends on the time increment At because the finite difference approximations become more accurate as At gets smaller. A summary of some iteration results and a comparison between this technique and the numerical integration with Gear s method will be presented after the following discussion on the stability of the temperature equation. 

The finite difference approximations can either be applied to the derivatives on the line from which the solution is advancing or on the line to which it is advancing, the former giving an explicit finite difference scheme and the latter an implicit scheme. The type of solution procedure obtained with the two schemes is illustrated in Fig. 3.18. 

Although not necessary, it is convenient, in most cases, to write these equations in dimensionless form before deriving the finite difference approximations to them. For the case of laminar flow here being considered, the following dimensionless variables are convenient to use ... 

Consider first the finite difference approximation for dU/dY at the point i, j. In order to derive this finite-difference approximation it is noted that the values of U at... 

If the finite difference approximations given in Eqs. (3.183), (3.186), and (3.187) are substituted into the momentum equation, an equation that has the following form is obtained on rearrangement ... 

The finite-difference approximation for dUldX given in Eq. (3.184) is used to determine the right-hand side of this equation. Therefore, since the continuity equation can be written as ... 

Using the values of F at the five nodal points shown in Fig. 4.10, the following finite-difference form of Eq. (4.106) is obtained using the finite-difference approximations introduced in the previous chapter in the discussion of the numerical solution of external flows ... 

The continuity equation (6.127) has exactly the same form as that for laminar flow and it can therefore be treated using the same procedure as used in laminar flow. The grid points shown in Fig. 6.10 are therefore used in obtaining the finite difference approximation to the continuity equation. 

Once the dimensionless mixing length distribution has been found using the above equations, the dimensionless eddy viscosity distribution, in terms of which the numerical method has been presented, can be found using the finite-difference approximation for the velocity derivative that was introduced earlier, i.e., using ... 

Consider the two points adjacent to the surface AB that are shown in Fig. 8.27. A Taylor expansion gives to the same order of accuracy as used in obtaining the finite-difference approximations used in the rest of the analysis. 

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