Two-Point First Derivative Approximations

The Finite Differenee Method ean be used to approximate eaeh term in this equation by using the differenee equation for the first partial derivative. The values of the funetion at two points either side of the point of interest, k, are determined, and 1. These are equally spaeed by an inerement Ax. The finite differenee equation approximates the value of the partial derivative by taking the differenee of these values and dividing by the inerement range. The terms subseripted by indieate... 

From Eq. (1.14), it is possible to obtain different approximations for the first and second derivatives of a given function. For example, it is possible to approximate the first derivative with a two-point expression ... 

Linear interpolation, or first order interpolation, is the simplest form of solving the problem, since it assumes a linear behavior between two adjacent points this is clearly depicted in Fig. 7.1. Here, we only use information given by the two points. Often, this is a good approximation, however, if a better interpolation is desired, we need to include data points beyond the two points used to interpolate. The inclusion of more points increases the order, and may include derivatives of the function at those points. 

We now have three two-point approximations for a first derivative, all in fact being the same expression, (y2 — Vi)/h, but depending on where this formula is intended to apply, being, respectively a forward difference of 0(h) if applied at xt, a backward difference of 0(h) if applied at x2 and a central difference of 0(h2) if applied at (.iq +. r2)/2. In subsequent chapters, all these will be used to approximate, among others (2.3)-(2.8). 

The above approximations to a first derivative used only two points, which sets a limit on the approximation order. By using more points, higher-order approximations can be achieved. In the context of this book, forward and backward multi-point formulae are of special interest, as well as some asymmetric and centra] multi-point ones. To this end, a notation will be defined here. Figure 3.2 shows the same curve as Fig. 3.1 but now seven points are marked on it. The notation to be used is as follows. If a derivative is approximated using the n values yi. . yn, lying at the x-values. iq. ..xn (intervals h) and applied at the point (Xi,yi), then it will be denoted as y (n) (for a first derivative) and y"(n) (for a second derivative). 

For those who prefer to keep the derivative approximation of G down to the two-point form, the above can perhaps be simplified a little the u-v device is not needed as such, as only the first substitution (6.6) is required. 

Unless direct methods are used to locate heavy atom positions, an understanding of the Patterson function is usually essential to a full three-dimensional structure analysis. Interpretation of a Patterson map has been one of two points in a structure determination where the investigator must intervene with skill and experience, judge, and interpret the results. The other has been the interpretation of the electron density map in terms of the molecule. Interpretation of a Patterson function, which is a kind of three-dimensional puzzle, has in most instances been the crucial make or break step in a structure determination. Although it need not be performed for every isomorphous or anomalous derivative used (a difference Fourier synthesis using approximate phases will later substitute see Chapter 10), a successful application is demanded for at least the first one or two heavy atom derivatives. 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

In Fig. 11.5.C-2 the locus of the partial pressure and temperature in the maximum of the temperature profile and the locus of the inflection points before the hot spot are shown as p and (pj), respectively. Two criteria were derived from this. The first criterion is based on the observation that extreme sensitivity is found for trajectories—the p-T relations in the reactor—intersecting the maxima curve p beyond its maximum. Therefore, the trajectory going through the maximum of the p -curve is considered as critical. This is a criterion for runaway based on an intrinsic property of the system, not on an arbitrarily limited temperature increase. The second criterion states that runaway will occur when a trajectory intersects (Pi)i, which is the locus of inflection points arising before the maximum. Therefore, the critical trajectory is tangent to the (pi)i-curve. A more convenient version of this criterion is based on an approximation for this locus represented by p in... 

Equation (4.1.15) implies that at any point in the medium there are two linear vibrations polarized along the local principal axes. The polarization directions of these two vibrations rotate with the principal axes as they travel along the axis of twist and the phase difference between them is the same as that in the untwisted medium. This result was first derived by Mauguin and is sometimes referred to as the adiabatic approximation. It is this property that is made use of in the twisted nematic device discussed in 3.4.2. 

If the first derivative is approximated with the most accurate formula that uses all three points, in practice only the two extreme points are used and the central point is skipped. In case of very large f ( ), the information to properly estimate the value of y at the central point is lost For this reason, all the programs based on the method of finite differences approximate the first derivative using a formula... 

Such an approximation is attractive for two reasons first, we dispose the (approximate) values of the potential energy for all points in the configuration space (not only those for which the calculations were performed), and second, the analytical formula may be differentiated and the derivatives give the forces acting on the atoms. 

---