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 [Pg.474]

Show that for any r and h a pure implicit difference scheme (a forward difference scheme) approximating the problem [Pg.380]

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. [Pg.160]

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

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 [Pg.520]

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. [Pg.266]

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. [Pg.66]

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 [Pg.229]

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 [Pg.497]

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. [Pg.224]

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. [Pg.222]

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 nonhomogeneities was employed with further reference to a priori estimates of z. The forward difference scheme with cr = 1 converges uniformly with the rate 0 h + r) due to the maximum principle. [Pg.495]

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 [Pg.5]

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 [Pg.348]

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. [Pg.316]

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 [Pg.321]

See also in sourсe #XX -- [ Pg.302 ]

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