Forward differences

The temporal derivative term in Equation (2.106) is approximated by a forward difference as... 

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. 

Calculate liquid densities, molar tray and condenser-reflux drum holdups, ana hquor and vapor enthalpies. Determine holdup and enthalpy derivatives with respect to time by forward difference approximations. 

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. 

More accurate difference expressions may be found by expanding the Taylor series. For example, f (x) to V(h) is given by forward difference by... 

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. 

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. 

Euler s method for solving the above set of ODEs uses a first-order, forward difference approximation in the -direction. Equation (8.16). Substituting this into Equation (8.21) and solving for the forward point gives... 

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

For a. first-order approximation, a straight line is fit between the points A = 0 and A to get the first-order, forward difference approximation... 

The third kind boundary conditions. The first kind boundary conditions we have considered so far are satisfied on a grid exactly. In Chapter 2 we have suggested one effective method, by means of which it is possible to approximate the third kind boundary condition for the forward difference scheme (a = 1) and the explicit scheme (cr = 0) and generate an approximation of 0 t -b h ). Here we will handle scheme (II) with weights, where cr is kept fixed. In preparation for this, the third kind boundary condition... 

This provides support for the view that the solution is completely distorted. From such reasoning it seems clear that asymptotic stability of a given scheme is intimately connected with its accuracy. When asymptotic stability is disturbed, accuracy losses may occur for large values of time. On the other hand, the forward difference scheme with cr = 1 is asymptotically stable for any r and its accuracy becomes worse with increasing tj, because its order in t is equal to 1. In practical implementations the further retention of a prescribed accuracy is possible to the same value for which the explicit scheme is applicable. Hence, it is not expedient to use the forward difference scheme for solving problem (1) on the large time intervals. 

Show that for any r and h a pure implicit difference scheme (a forward difference scheme) approximating the problem... 

This estimate can be improved for the forward difference scheme with (7 = 1 by means of the maximum principle and the method of extraction of stationary nonhomogeneities , what amounts to setting... 

Remark 2 Uniform convergence with the rate 0 h + r) of the forward difference scheme with cr = 1 can be established by means of the maximum principle and the reader is invited to carry out the necessary manipulations on his/her own. 

This case was studied by Nahavandi and Catanzaro (Nl) using an explicit forward difference formula with Ax set equal to the pipe length and the time step dictated by stability considerations. 

Let fix be defined for discrete equidistant values of x, which will be denoted by v,. The corresponding value of y will be written t/ = flxfj- The first forward difference () /(.v) denoted by Aflx) fix + h) — fix) where h = x — x , = interval length. 

Equally Spaced Forward Differences If the ordinates are equally spaced, i.e., x, — x . = Ax for all j, then the first differences... 

Central differences were used in Equation (5.8), but forward differences or any other difference scheme would suffice as long as the step size h is selected to match the difference formula and the computer (machine) precision with which the calculations are to be executed. The main disadvantage is the error introduced by the finite differencing. 

Finite difference Newton method. Application of Equation (5.8) to/(jc) = x2 - x is illustrated here. However, we use a forward difference formula for f x) and a three-point central difference formula for/"(jc)... 

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. 

One at a time, perturb the elements of the tear variable xTi. Calculate the dependent variables, and evaluate the tear equations. Calculate the gradients of/, g, h, and h with respect to each xT i by a forward difference equation in which the xD are the perturbed values and x7 are the unperturbed values. 

It is instructive to note that the standard first-order forward difference discretization of Eq. (50) is finite difference calculus ... 

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