Difference scheme forward

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. 

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. 

Other ideas are connected with two types of purely implicit difference schemes (the forward ones with cr = 1) available for the simplest quasi-linear heat conduction equation... 

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... 

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. 

We give a brief survey afforded by the above results scheme (II) converges uniformly with the same rate as in the grid L2(u>h)-norm (see (35)) if and only if condition (39) holds. The stability condition (39) in the space C for the explicit scheme with <7=0, namely r < h2, coincides with the stability condition (25) in the space L2(u)h) that we have established for the case it <. The forward difference scheme with a = 1 is absolutely stable in the space C. The symmetric difference scheme with cr = is stable in the space C under the constraint r < h2. 

Having no opportunity to touch upon this topic, we refer the readers to the aforementioned chapters of the manograph The Theory oof Difference Schemes , in which the method of extraction of stationary nonho-mogeneities was employed with further reference to a priori estimates of z. The forward difference scheme with c = 1 converges uniformly with the rate 0(h2 + r) due to the maximum principle. 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

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

O(At) signifies that in the above approximation the leading term that was neglected is of the order At (we have divided (8-6) by At to get (8-7)). This is the so-called Euler forward-difference scheme. While it is only first-order accurate in At, it has the advantage that it allows for the quantities at timestep n + l being calculated only from those known at timestep n. 

Since the solution of Equation (8-11) propagated at timestep tn+i is expressed solely in terms of data from timestep tn, not requiring any previous information, the forward-difference scheme is said to be explicit, and its essence can be extracted from Fig. 8-1, too. 

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 ... 

For fixed or chosen values of the parameters, the model equations (eqs. 1-4) along with the initial and boundary conditions (eqs. 5) are solved iteratively by a centered-in-space, forward-in-time, finite difference scheme to obtain (i) the hexene and hexene oligomer concentration profiles in the pore fluid phase, and (ii) the coke (extractable + consolidated) accumulation profde. The effectiveness factor (rj) is estimated from the hexene concentration profile as follows ... 

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 ... 

Equation (7) can most readily be solved by an explicit finite difference scheme which steps forward in time across the spatial grid. The value of G is updated at each spatial grid point in turn. When all of the spatial grid has been updated a solution at that point in time has been calculated for the problem considered. 

In principle, we can use any combination of forward, backward and center finite differences to replace the first two differential terms of Eq. 10.72. However, our choice is limited for two reasons. First, we need to achieve numerical stability of the solution (see Section 10.3.5.1 below). For example, the stability analysis shows that the finite difference scheme which replaces the first term of Eq. 10.72 by a... 

Equation 10.79 permits the calculation of the concentration at the new space position, n + 1, knowing the concentration at the previous space position, n (Godunov scheme). This method calculates band profiles along the coliunn at successive time intervals. The elution profile is the history of concentrations at z = I. The forward-backward difference scheme was first used to calculate solutions of the equilibrium-dispersive model of chromatography by Rouchon et al. [46,47]. Since then, it has been widely used. It is particularly attractive because of its fast execution by modern computers [50]. Czok [50] and Felinger [54] have shown how the CPU time required can be further shortened by eliminating the needless computation of concentrations below a certain threshold. The dramatic increase over the last fifteen years of the speed of the computers available for the numerical calculations of band profiles has considerably reduced this advantage of the Forward-Backward scheme over the other possible ones. 

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 ... 

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