Backward Finite Differences

Higher-order backward differences are similarly derived  

The coefficients of the terms in each of the above finite differences correspond to those of the binomial expansion (a - b) , where n is the order of the finite difference. Therefore, the general formula of the nth-order backward finite difference can be expressed as 

It should also be noted that the sum of the coefficients of the binomial expansion is always equal to zero. This can be used as a check to ensure that higher-order differences have been expanded correctiy. 

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 

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Here, we obtained a backward finite difference expression for the derivative d/dx at a point i, and we get the order of the interpolation, i.e., all the terms that we collapsed in 0(Ax), which is a metric for the error built into the approximation. Similarly, we can obtain the forward difference approximation as... 

The boundary condition at x = L or n = 5 has a first derivative that comes from the balance between the conduction and the convection at the end of the fin. We can use a backward finite difference to approximate this derivative, i.e.,... 

By using a backward finite differences, essentially up-winding the convective term, we get... 

To obtain an algorithm that is unconditionally stable, we consider an implicit discretization scheme that results from using backward finite-differences for the time derivative. The corresponding difference equation is most conveniently obtained by approximating the diffusion equation at point (Xj,tn+i) ... 

Moreover, applying the backward finite difference, yields... 

In this woric, discretisation of both space and time derivatives was executed, based on either central finite difference (CFD) or orthogonal collocation cm finite elements (OCFE) discretisation in the spatial domain and backward finite difference (BFD) discretisation in the time domain. 

With previously published kinetic constants [10,11], presented in Table 1, the model was solved for two different initial conditions. The former one (type I) presumes the existence of a reactants and products profile at t=0, whereas the latter (type II) considers that the reactor is empty at t=0. The resultant differential-algebraic system of equations was solved by backward finite differences formula with variable step, implemented in the DASSL code [12,13], The numerical convergence was assured by increasing the number of finite elements until no further modification in the model simulations was obtained. Hence, the number of elements was gradually increased fi om 20 to 100. Nevertheless, no significant diference was observed, i. e., all product yields were obtained with errors smaller than 10 . ... 

Thus, when we replace a partial differential term by a forward or a backward finite difference, we make an error which is of the same order as 0 Ax), and the coefficient of this error contribution is the second-order partial differential, G/dx. ... 

In conclusion, when we replace the first- and second-order partial differential terms in a partial differential equation by central finite difference terms, we make errors that are of the order of 0 Ax ). For most practical purposes, this second-order error is negfigible. By contrast, when we replace the first-order partial differential terms with a forward or a backward finite difference term, we make errors that are of the order of 0 Ax). This first-order error contribution is never negligible. 

As an example, if we select a central finite difference for the first term in the LHS of Eq. 10.61, a backward finite difference term for its second LHS term, and a... 

In this calculation scheme, the first term of Eq. 10.72 is replaced by a forward finite difference while the second term is replaced by a backward finite difference. We obtain the equation... 

The backward-forward finite difference scheme is identical to the Craig model if we choose the time and space increments such that = H. The Craig model has been used by many authors, including Eble et ah [45], Czok and Guiochon [49,50], and El Fallah and Guiochon [55]. This model affords a good numerical solution of the gradient elution problem, which is very difficult to solve numerically with the forward-backward finite difference scheme [55,56]. 

With the forward-backward finite difference scheme, the space increment, h, and the time increment, t, are chosen such that (fl — 1) — Da/u. Since D = Hu/2, where H is the column HETP, we obtain for the first scheme the following condition ... 

With the forward-backward finite difference scheme ... 

There is an abimdant literature on the comparison between experimental and calculated band profiles for binary mixtures. The most popular methods used have been the forward-backward finite difference scheme and the OCFE method. The former lends itself readily to numerical calculations in many cases representative of the present preoccupations in preparative chromatography. We present first a comparison between the band profiles obtained with the ideal and the equilibrium-dispersive model to illustrate the dispersive influence of the column efficiency. Related to the comparison between these two models is the issue of the use of the hodograph transform of experimental results discussed in Section 11.2.2. Computer experiments are easy to carry out and most instructive because it is possible to show e effects of the change of a single parameter at a time. Some... 

Figure 11.6 Comparison of profiles calculated with OCFE and a finite difference method. N = 1000 plates. Line 1, forward-backward finite difference algorithm, Courant number = 2 line 2, OCFE, fp = 5 s. (a) 1 9 mixture, a = 1.5. (b) 1 1 mixture, a. = 1.5. (c) 5 1 mixture, a = 1.2.

When backward finite difference accurate to the order h is used for the first derivative in the governing equation, the solution does not oscillate, and the following plots are obtained for N = 10 node points. 

For these simulations, the discretization of the mass balance equations used first order backward finite differences over a uniform grid of 100 intervals in the axial direction and a third order orthogonal collocation over 50 finite elements in the radial direction [16]. 

Finite difference formulations may occur as any one of three types, namely forward, central, or backward finite difference [5,9,25]. Generally, these formulations lead to nonlinear systems of equations. The methods and approaches discussed in Section 9.2 can be employed. However, if the resulting system of equations is linear, then the methods of Section 9.3 apply. Next, we will briefly discuss a linear central difference and a nonlinear central difference formulation. 

The set of PDEs (10.39)-(10.46), with obvious initial and boundary conditions was solved numerically according to the method of lines, based on axial discretization with backward finite differences and on time integration by Gear s algorithm. 

A simple digital form of this equation may be written for the nth time interval using the rectangular rule of integration approximation to the integral and a first-order backward finite difference approximation of the derivative to yield the positional form, Eq. (92), where are the error and controller output respectively... 

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. 

First-Order Derivative in Terms of Backward Finite Differences with Error of Order h... 

Eq. (4.16), therefore, enables us to evaluate the first-order derivative of, v at position i in terms of backward finite differences. 

This equation evaluates the second-order derivative of y at position 1, in terms of backward finite differences, with error of order h. 

Table 4.1 Derivatives in terms of backward finite differences...

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