Time-step size

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

For practical purposes, implicit schemes are the methods of choice when the solution is smooth and well behaved as a function of time. In that case much larger time steps can be taken than with explicit schemes, thus allowing a reduction in the computational effort. When large temporal gradients and rapid variations are expected, accuracy constraints set severe limits to the time-step size. In that case explicit schemes might be favorable, as they come with a reduced numerical effort per time step. 

In contrast to the method of characteristics, which gives faithful simulation of transient flows but which is very restrictive in time step sizes, the stability of the implicit methods permit large time steps and drastic reduc-... 

If the most recent available measurements are at time step c, then a history horizon HAt can be defined from (tc — HAt) to tc, where At is the time step size. In order to obtain enough redundant information about the process, it is important to choose a horizon length appropriate to the dynamic of the specific system (Liebman et al., 1992). As shown in Fig. 5, only data measurements within the horizon will be reconciled during the nonlinear dynamic data reconciliation run. 

Table 3.7 compares the accuracy for two different time step sizes. Obviously, final accuracy depends on the time step chosen and considerable computational effort would be required for a good approximation. 

Dt = 1E10 initialize time step size to a large number... 

Develop a set of initial total flow rates and temperatures. Also set liquid compositions for trial k = 1 and the time step size, At. 

This establishes the upper limit on the solids stream integration time step size for any specified grid spacing. Likewise, for the time method of lines model, the stability requirement was estimated as... 

The round-off error is proportional to the number of computations performed during the solution. In the finite difference metliod, the number of calculations increases as the mesh size or the time step size decreases. Halving the mesh or time step size, for example, doubles the number of calculations and thus the accumulated round-off error. 

As the mesh or time step. size decreases, the discretization error decreases but the round-off error increases. 

There are three major problems for the Euler method. First, the accuracy is poor, since the method is based upon Eq. (F.16), in which only a first-order difference expression is used. The errors in the method are proportional to At. Second, stability is difficult to achieve for many problems. The only way to have a stable Euler method is to use a small enough time step-size, but you may not know what value is sufficient. Furthermore, a value that is sufficient at the beginning may not be sufficient later on, and it may take an excessively long time to finish the computation. Third, to validate the results it is necessary to solve the problem at least twice, with different time-steps. The method can, however, be programmed in Excel, as Figures A.3 and A.4 demonstrate. 

In order to obtain confidence in the results of numerical simulations, numerical experiments are performed. These involve solving similar problems, for which theoretical or numerical treatments or high-quality experimental data are available. Ultimately, however, simulations of the problem of interest must be verified by increasing the grid density until the solution does not change when additional grid points are added. For an unsteady solution, convergence on time-step size must also be verified. 

It becomes apparent that such spatial operators take into account both the time-step size and the shape of the elementary cells, namely they depend on the anisotropic features of the FDTD simulation. Besides, as compared to the schemes of Section 2.5.4, they exhibit a more narrow-band performance. 

Now, consider an instant i where the solution is known and introduce a time-step size... 

Table 1 A summary of XRD data for the Ce doped perovskite system, after aging at 800°C for 12 hours. Phase amounts, as calculated by Siroquant are given. However these can only be approximate values since data collection time, step sizes etc. were too small for very accurate calculations. Also included are BET surface areas for the doped perovskites.

Due to this limitation of straightforward MD algorithms, numerous approaches have been developed. Some of them have aimed to increase the time step size [4-15]. Typical improvements in this multiple steps algorithms are rather modest (one order of magnitude) so they do not provide a satisfactory solution to the timescale problem. 

With models for catalyst decay and effectiveness now in hand, the simulation of lignin liquefaction could be achieved given the initial lignin structure (as described earlier) and model compound reaction pathways and kinetics, both thermal and catalytic. Construction of a random polymer, as outlined earlier, began the simulation. This structural information combined with the simulated process conditions to allow calculation of the reaction rate constants, selectivities and associated transition probabilities. The largest rate constant then specified the upper limit of the reaction time step size. 

Kuljanic et al. (2007) take up this deficit by adapting the time step size to signal changes of the cutting force coefficients or the cutting depth. Furthermore, these time steps were no longer assumed to be constant but are interpolated by polynomials within their range (see Fig. 8). 

Thus, we see that this formula is basically the Crank-Nicolson formula written for the time step size of 2(At) (compare this with Eq. 12.155 where the step size is Af). 

Figures 12.11 shows plots of yj = y x = 0.2) as a function of time. Computations from the forward difference scheme are shown in Fig. 12.11a, while those of the backward difference and the Crank-Nicolson schemes are shown in Figs. 12.11h and c, respectively. Time step sizes of 0.01 and 0.05 are used as parameters in these three figures. The exact solution (Eq. 12.129) is also shown in these figures as dashed lines. It is seen that the backward difference and the Crank-Nicolson methods are stable no matter what step size is used, whereas the forward difference scheme becomes unstable when the stability criterion of Eq. 12.139 is violated. With the grid size of Aa = 0.2, the maximum time step size for stability of the forward difference method is At = (Ajc)V2 = 0.02.

Fread, D.L. (1973). Effects of time step size in implicit dynamic routing. Water Resources Bulletin 9(2)-. 338-351. 

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