Evaluation explicit method

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

In an explicit method, the spatial derivatives are evaluated based on the dependent variables at time level n, that is, w". Thus, for the problem at hand,... 

Compared to the explicit Euler method (Eq. 15.9), note that the right-hand side is evaluated at the advanced time level tn+1- If f(t, ) is nonlinear then Eq. 15.22 must be solved iteratively to determine yn+. Despite this complication, the benefit of the implicit method lies in its excellent stability properties. The lower panel of Fig. 15.2 illustrates a graphical construction of the method. Note that the slope of the straight line between y +i and yn is tangent to the nearby solution at tn+, whereas in the explicit method (center panel) the slope is tangent to the nearby solution at t . 

An alternative, called semi-implicit methods in such texts as [351], avoids the problems, and some of the variants are L-stable (see Chap. 14 for an explanation of this term), a desirable property. This was devised by Rosenbrock in 1962 474]. There are two strong points about this set of formulae. One is that the constants in the implicit set of equations for the k s are chosen such that each can be evaluated explicitly by easy rearrangement of each equation. The other is that the method lends itself ideally to nonlinear functions, not requiring iteration, because it is, in a sense, already built-in. This is explained below. 

The same approach may be used in conjunction with the linearized equations and explicit methods of Section 8.5. Basically, [A ] is calculated from Eq. 8.3.25 using the average mole fractions and the binary i — j pair mass transfer coefficients in the evaluation of [7 ]. 

A number of investigators used the wetted-wall column data of Modine to test multicomponent mass transfer models (Krishna, 1979, 1981 Furno et al., 1986 Bandrowski and Kubaczka, 1991). Krishna (1979b, 1981a) tested the Krishna-Standart (1976) multicomponent film model and also the linearized theory of Toor (1964) and Stewart and Prober (1964). Furno et al. (1986) used the same data to evaluate the turbulent eddy diffusion model of Chapter 10 (see Example 11.5.3) as well as the explicit methods of Section 8.5. Bandrowski and Kubaczka (1991) evaluated a more complicated method based on the development in Section 8.3.5. The results shown here are from Furno et al. (1986). 

For the remaining VPVP a) correction, values for a few Z were presented by Manakov and Nekipelov [58]. We present our method of calculation here. Instead of evaluating explicitely the corresponding energy shift,... 

The method of solution proposed uses rate evaluations at (z, y) to determine new profiles at (z +1, j). For a fine z grid this will present no problem since only small changes in the dependent variables with Az are involved. However, problems with the stability of the solution may arise if the grid is too course. It has been shown that this explicit method is stable so long as... 

A method that does not need any evaluation of f in y +j, tn+i is called an explicit method. 

In the current terminology for multistep methods, the use of the explicit method is denoted as P (prediction), the calculation of the functions f with E (evaluation), and the correction obtained by means of an implicit method with C (correction). 

We have presented the essence of the Runge-Kutta scheme, and now it is possible to generalize the scheme to a pth order. The Runge-Kutta methods are explicit methods, and they involve the evaluation of derivatives at various... 

FIGURE 5.3. The model used in the two-region calculation of defect energies. In Region I, which contain a few hundred atoms forming the immediate environment of the defect d, interactions are evaluated explicitly. Between this and the outer Region II, whose contributions are treated by bulk dielectric methods, lies the interface Region Ila. 

Some of the discrepancies between the Wiepking-Doyle data and the Soden-McLeish data may have been systematic. The Wiepking-Doyle (linear) results were based on a sample population that uniformly had a 12 percent moisture content. The Soden-McLeish (exponential) results,however, were derived in part from the Doyle et al. work based on samples with 9.5 percent moisture content, and in part on their own experiments for which no moisture contents were reported. It is difficult to tell whether these variations in percent moisture could have biased the results significantly. No moisture-related effects on the properties of balsa wood have been evaluated explicitly in any of the literature references surveyed here. To the extent that moisture contents were reported at all, they ranged generally from 8 to 12 percent. This range appears to be a normal condition for kiln-dried balsa wood exposed to ambient atmosphere in temperate climates. (Note The ASTM Standard Method of Testing Small Clear Specimens of... 

Ab initio calculations require a great deal of computer time and memory, because every term in the calculations is evaluated explicitly. Semiempirical calculations have more modest computer requirements, allowing the calculations to be completed in a shorter time and making it possible to treat larger molecules. Chemists generally use semiempirical methods whenever possible, but it is useful to understand both methods when solving a problem. 

Equations (4) and (5) are not evaluated explicitly in the minimization program, but are fit using a combination of spline [17] methods, which provide stability, the ability to filter noise easily, and the flexibility to describe an arbitrarily shaped potential curve. Moreover, the final functional form is inexpensive to evaluate, making it amenable to global minimization. The initial step in our methodology is to fit the statistical pair data for each amino acid and for the density profile to Bezier splines [17]. In contrast to local representations such as cubic splines, the Bezier spline imposes global as well as local smoothness and hence effectively eliminates the random oscillatory behavior observed in our data. 

In this expression, ( ) is the solution at time t and is the solution at time t + At. While certain flow conditions, such as compressible flow, are best suited to an explicit method for the solution of eq. (5-38), an impficit method is usually the most robust and stable choice for a wide variety of applications, including mixing. The major difference between the explicit and implicit methods is whether the right-hand side of eq. (5-38) is evaluated at the current time [F(( )) = F(ct))"] or at the new time [F(c )) = F(( )"+ )]. The implicit method uses the latter ... 

Obviously, the PRE diminishes when a value of At is sufficiently small. Hence, the smaller the At, the higher the accuracy of the approximate solution. On the other hand, the number of steps and function evaluations will increase as At is reduced, which means a longer computational (CPU) time. There is a trade-off between these two extremities. Table 7.1 compares Euler s explicit method with the analytical method at each step of integration, using the PRE as the indicator of goodness. 

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