Forward-backward differences

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Coefficients of forward difference expressions for derivatives of up to the fourth order are given in Figure 1-52 and of backward difference expressions in Figure 1-53. 

Given a data table with evenly spaced values of x, and rescaling x so that h = one unit, forward differences are usually used to find f(x) at x near the top of the table and backward differences at x near the bottom. Interpolation near the center of the set is best accomplished with central differences. 

When the user, whether working on stand-alone software or through a spreadsheet, supplies only the values of the problem functions at a proposed point, the NLP code computes the first partial derivatives by finite differences. Each function is evaluated at a base point and then at a perturbed point. The difference between the function values is then divided by the perturbation distance to obtain an approximation of the first derivative at the base point. If the perturbation is in the positive direction from the base point, we call the resulting approximation a forward difference approximation. For highly nonlinear functions, accuracy in the values of derivatives may be improved by using central differences here, the base point is perturbed both forward and backward, and the derivative approximation is formed from the difference of the function values at those points. The price for this increased accuracy is that central differences require twice as many function evaluations of forward differences. If the functions are inexpensive to evaluate, the additional effort may be modest, but for large problems with complex functions, the use of central differences may dramatically increase solution times. Most NLP codes possess options that enable the user to specify the use of central differences. Some codes attempt to assess derivative accuracy as the solution progresses and switch to central differences automatically if the switch seems warranted. 

Figure 8.2 Schematic of the backward, forward and central difference schemes.

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... 

Note that due to the fact that the physical distance between the grid points, Ax, is small (Ax -Cl), and that the central difference is of second order ( Ax2) nature, the central difference approximation leads to a much better solution than the first order backward and forward difference solutions. 

The species balance relation Eq. 13.2-8 is transformed to a difference equation using the forward difference on the time derivative and the backward difference on the space derivative. The finite difference form of the x-momentum equation (Eq. 13.2-25) is obtained by using the forward difference on all derivatives, and is solved by the Crank-Nicolson method. The same is true for the energy equation (Eq. 13.2-26). 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

Figure8-1 Space-time grid for the one-dimensional diffusion equation, evidencing the explicit forward-difference, implicit backward-difference and C rank-Nicholson discretization schemes.

The implicit backward-difference algorithm does not show the stability problems encountered in the case of the explicit forward-difference method, and this results immediately by analyzing the expression of the amplification factor ... 

This is the so-called Crank-Nicholson scheme and, formally, it could have been obtained by simply averaging the explicit forward-difference and implicit backward-difference schemes. By conveniently grouping the terms, the following system of linear equations results ... 

The following are examples of the different identifying sentence errors questions that you will be tested on in the SAT. There are 114 practice questions, which means that you will know these types of questions backward, forward, and upside down ... 

In this explicit scheme, the first-order forward difference approximation is used for the time derivative. The second-order central difference approximation is used for the spatial derivatives. When a first-order backward difference approximation (c > 0) for the convective term is used, then the FDE of the PDE Eq. (10.2) is... 

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). 

In this way, the coefficients for any y((n) can be calculated. Table A.l in Appendix A shows them all, as whole numbers m/3j, where m is the multiplier mentioned above. For each n, the Table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. For n up to 6, all possible forms are included, as they will be needed later, while for n = 7, only the forward and backward formulae are shown, as only these are needed. In case the reader wonders why all this is of interest the forms y[(n) will be used to approximate the current in simulations (see the next section) the backward forms y n(n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the Table will be used for the Kimble White (high-order) start of the BDF method, also described in these chapters. The coefficients have a long history. Collatz [169] derived some of them in 1935 and presents more of them in [170]. Bickley tabulated a number of them in 1941 [88], The three-point current approximation, essentially y((3) in the present notation, was first used in electrochemistry by Randles [460] (preempted by two years by Eyres et al. [225] for heat flow simulations), then by Heinze et al. [301], and schemes of up to seven-point were provided in [133]. 

Equation (4-41) is developed by using the forward-difference concept to produce an explicit relation for each Tf+l. As in our previous discussion, we could also write the energy balance using backward differences, with the heat transfers into each ith node calculated in terms of the temperatures at the p + 1 time increment. Thus,... 

Li, S. C., and Lewandowsky, S. (1995). Forward and backward recall different retrieval processes. Journal of Experimental Psychology Learning, Memory, and Cognition, 21 (4), 837-847. 

Bilateral symmetry is very common in the animal kingdom (Figure 2-3). It always appears when up and down as well as forward and backward are different, whereas left-bound and right-bound motion have the same probability. As translational motion along a straight line is the most characteristic for the vast majority of animals on Earth, their bilateral symmetry is trivial. This symmetry is characterized by a reflection plane, or mirror plane, and its usual label is m. 

Working in the same way, with different backward-forward finite difference schemes for the second-order finite differences, the solution of Equation 11.6 is as follows ... 

If the reaction rate constants for the forward and backward reactions differ considerably, the reaction time is dominated by the larger rate constant. The... 

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. 

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 ... 

We use frequently the forward-backward scheme with a = 2 and h = H, which satisfies Eq. 10.96 [4 50]. If calculations are made with this scheme and a= with e very small, of the order of 1 x 10 , the numerical solution obtained is practically identical to the solution of the ideal model (H = 0), as expected. Czok et al. used the backward-forward scheme with different values of a [49,50]. Lin... 

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