Finite-Difference Approximations

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d2f /dx2. The basic idea is to approximate/as a polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df/dx and d2f/dx2. The polynomial approximation is a local one that applies to some region of space centered about point v. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

This is a second-order approximation and can be used to obtain derivatives up to the second. Differentiate to obtain 

The value of the first derivative depends on the position at which it is evaluated. Setting x = +Ax gives a second-order, forward difference.  

Setting x = 0 gives a second-order, central difference.  

Setting x = Ax gives a second-order, backward difference.  

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With derivatives with respect to temperature and composition of necessity found by finite difference approximation. 

In the case of the adiabatic flash, application of a two-dimensional Newton-Raphson iteration to the objective functions represented by Equations (7-13) and (7-14), with Q/F = 0, is used to provide new estimates of a and T simultaneously. The derivatives with respect to a in the Jacobian matrix are found analytically while those with respect to T are found by finite-difference approximation... 

Value of the objective function [(7-23) or (7-24)] at T + AT used for finite difference approximation of the derivative. 

An alternative expression is based on the following finite difference approximation [Brunger et al. 1984] ... 

Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonhnear processes. If dcjdi on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets ... 

Divide the tube length into a number of equally sized increments, = LjJ, where J is an integer. A finite difference approximation for the partial derivative of concentration in the axial direction is... 

This approximation is called a forward difference since it involves the forward point, z + Az, as well as the central point, z. (See Appendix 8.2 for a discussion of finite difference approximations.) Equation (8.16) is the simplest finite difference approximation for a first derivative. 

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. 

Apply finite difference approximations to Equation (9.15) using a backwards difference for da/d and a central difference for d a/d. The result is... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

These relations may be utilized to calculate ASV, or AU() >i in a finite-difference approximation... 

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 55-7 First derivatives calculated using different spacings for finite difference approximation to the true derivative. The underlying curve is the 20 run bandwidth absorbance band in Figure 54-1, with data points every nm. Figure 55-7a Difference spacings = 1-5 nm Figure 55-7b Spacings = 5 10 run Figure 55-7c Spacings = 40-90 nm. (see Color Plate 21)...

By replacing the derivatives in the transport equation with finite difference approximations, we introduce to our numerical solution specific inaccuracies. To see this, we write a Taylor series expanding the difference between and... 

Comparing this equation to a finite difference approximation (Eqn. 20.29), we see that in the numerical solution we carry only the first term in the series, the d Q /3x term, omitting the higher order entries. The Taylor series is truncated, then, and the resulting error called truncation error. 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

From numerous tests involving optimization of nonlinear functions, methods that use derivatives have been demonstrated to be more efficient than those that do not. By replacing analytical derivatives with their finite difference substitutes, you can avoid having to code formulas for derivatives. Procedures that use second-order information are more accurate and require fewer iterations than those that use only first-order information(gradients), but keep in mind that usually the second-order information may be only approximate as it is based not on second derivatives themselves but their finite difference approximations. 

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