Forward differencing

Determine F(t), Eft), t, and of for flow through the vessel, using both backward and forward differencing for E(t), and calculating t and of from Eft)... 

In forward differencing we let 5V, = V, — Vi i in backward differencing we let 5Vi- = Vi — Vi. In central differencing we say that a particular difference is associated with the interval between two of the original values, and that therefore <5V 1/2 K — K i While this is less convenient for computation (how many programming languages allow for half-integer valued subscripts ), it reflects the semantics of the sequence better, and we will use it as our convention. 

Equation 15 was used as a constraint with a value between 12 and 13 for Z (n-decane conversion), during optimization of the reaction variables, using a Non-linear Quasi-Newton search method with tangential extrapolation for estimates, forward differencing for estimation of partial derivatives, a tolerance of 0.05 and precision of 0.0005. The search was also constrained by boundary conditions 1 to -1 for the reaction variables x, and solved for maximization of Y . 

At each time step, values of the variables are calculated at every i, and these are used as input for the next time stepping. This scheme of discretization is known as Lax-Friedrichs finite difference scheme, which is first order accurate [22,23]. In order to ensure stability during time stepping, the variables at time n are approximated as the average of their values at (i - l)th and (f+l)th nodes instead of simple forward differencing, that is, (cm)" -(cm) or According to the Courant-... 

Elliott, S., Turco, R.P., Jacobson, M.Z. Tests on combined projection forward differencing integration for stiff photochemical family systems at long-time step. Comput. Chem. 17, 91-102 (1993)... 

One important conclusion can be drawn from this simple example. The forward difference technique has severe problems with differential equations that have differing time constants. Also as shown by diese examples, the accuracy does not approach that of the trapezoidal or backwards differencing rule. Thus the forward differencing algorithm will be eliminated from further consideration as a general purpose technique for the numerical solution of differential equations. 

For the time derivative there are several possible approximations. Three of these have been discussed in detail in Chapter 10 and are known as flie explicit forward differencing (FD) method, flie imphcit backwards differencing (BD) method and the trapezoidal rule (TP) which averages flie time derivative between two successive time points. From the discussion of fliese methods in Section 10.1, one would expect different long term stability results for each of fliese methods and this is certainly the case for partial differential equations as well as single variable differential equations. The forward and backwards time differencing methods leads to the set of equations ... 

As shown in this chapter, in the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaeed by their finite-differenced equivalents. These finite-differenced forms of the model equations are shown to evolve as a natural eonsequence of the balance equations, according to the manner of Franks (1967). The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end sections can... 

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. 

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... 

The significance of this result is that the timestep At insuring the stability of the algorithm is limited by an upper bound, which is proportional to the diffusion time across a cell of width h. This makes the explicit scheme, characterized by forward time differencing, conditionally stable and proves that the value A - 1/2 is indeed critical. 

Maudsley (1984) proposes central differences for the inner points, with forward and backward differences at the boundaries. The pipeline is split into K pipe reaches, each of which is assumed to have two state variables associated with it a pressure and a flow, both of which will vary with time. Central differencing... 

An alternative differencing scheme may be arrived at by considering the average derivative given by the forward and backward schemes this is called the central difference approximation ... 

To facilitate the development of explicit and implicit methods, it is necessary to briefly consider the origins of interpolation and quadrature formulas (i.e., numerical approximation to integration). There are essentially two methods for performing the differencing operation (as a means to approximate differentiation) one is the forward difference, and the other is the backward difference. Only the backward difference is of use in the development of eiqjlicit and implicit methods. 

F/9x. Our use of left and right values to define geometric slopes (for both first and second derivatives) is called central differencing. Backward and forward one-sided differencing are also possible, though less accurate for the same number of points. 

The other extreme is to evaluate the remaining terms at time n + 1)((50 the fully implicit or backward differencing approach. It leads to a set of algebraic equations from which the dependent variables at time (n -h 1)( 0 can be calculated. This approach is unconditionally stable (Richtmyer and Morton, 1967), and is the approach used here. We may of course also use other schemes in which intermediate weights are given to the forward and backward differences. These partially implicit schemes lead to improved accuracy. However, if attempts are made to use them on systems of stiff equations, the latter must be treated by asymptotic techniques. In chemical situations such techniques are equivalent to the use of the chemical quasi-steady-state or partial equilibrium assumptions at long times. They will be considered again in Section 9. 

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