Taylor explicit method

The great advantage of the methods described in this section over those described earlier is, of course, rapidity in computation. This gain in computational simplicity is, however, at the expense of theoretical rigor. It is, therefore, important to establish the accuracy of the methods described above using the exact method of Section 8.3 as a basis for comparison. The extensive numerical computations made by Smith and Taylor (1983) showed that the explicit method of Taylor and Smith ranked second overall among seven approximate methods tested (the linearized method of Section 8.4 was best). For some determinacy... 

The effective diffusivity formula of Stewart (Eq. 6.1.8) is by far the best of this class of methods. This should not come as a surprise since this method is capable of correctly identifying the various interaction phenomena possible in multicomponent systems. Indeed, for equimolar countertransfer, this effective diffusivity method is equivalent to the linearized theory and to both explicit methods discussed above. In fact, for some systems Stewart s effective diffusivity method is superior to Krishna s explicit method (Smith and Taylor, 1983). However, since the explicit methods are actually simpler to use than Stewart s effective diffusivity method (all methods require the same basic data) and, in general... 

Show that the explicit method of Taylor and Smith (1982) (discussed in Section 8.5) is an exact solution of the Maxwell-Stefan equations if all the binary diffusion coefficients are equal. Solutions are given by Burghardt (1984) and Taylor (1984). 

The explicit method of Taylor and Smith (1982) discussed in Section 8.5 makes use of the mass transfer rate factor... 

Sometimes the ODEs that arise in studies in nonlinear dynamics can be solved using explicit methods (such as the forward Euler) which require less computations per step and are thus cheaper and ter to implement. The Runge-Kutta femily of algorithms are a popular implementation of the explicit methods. Runge—Kutta methods begin with a Taylor series expansion the order of the particular Runge-Kutta method used is simply the highest order term retained in the Taylor series. 

Whether boxes or points are used, it is clear from the Taylor expansions (particularly) for the forward finite time differences expressing 9c/8t in discrete form, with an 0(St) error, that the explicit method does not converge very well. To some extent, this is no longer such a bad thing, since computers are everywhere and they are faster and cheaper. Many simulations are done on microcomputers and workers do not mind if a run takes all night. 

For stiff differential equations, an explicit method cannot be used to obtain a stable solution. To solve stiff systems, an unrealistically short step size h is required. On the other hand, the use of an implicit method requires an iterative solution of a nonlinear algebraic equation system, that is, a solution of k, from Equation A2.4. By a Taylor series development of y + /=i il i truncation after the first term, a semi-implicit Runge-Kutta method is obtained. The term k, can be calculated from [1]... 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

We have also studied the generalized Boltzmann operator for four particles in the Choh-Uhlenbeck formalism (see, for example, this expression in the work of Cohen8). Since the paper by Stecki and Taylor,29 it seems clear that the results of Bogolubov and those of Choh and Uhlenbeck (which are derived from them) must be equivalent to the Prigogine formalism. However, it is very difficult to show this equivalence explicitly in the concentration form that we have presented here. Indeed, in the Choh and Uhlenbeck method, the streaming operators appear with their Fourier components < k S(i n) k )and< k 0 S(i -n) k >, and the products of these operators are then expressed in terms of the Y(1 > for several complex variables. This renders the mathematical operations extremely complicated (see Eq. 111). 

To compare the explicit and implicit Euler methods we exploited that the solution (5.3) of (5.2) is known. We can, however, estimate the truncation error without such artificial information. Considering the truncated Taylor series of the solution, for the explicit Euler method (5.7) we have... 

Taylor et al. further advance the method by introducing explicitly a plane of ions with the countercharge rather than a homogeneous countercharge in the liquid phase.61 The electrochemical... 

The local composition model (LCM) is an excess Gibbs energy model for electrolyte systems from which activity coefficients can be derived. Chen and co-workers (17-19) presented the original LCM activity coefficient equations for binary and multicomponent systems. The LCM equations were subsequently modified (1, 2) and used in the ASPEN process simulator (Aspen Technology Inc.) as a means of handling chemical processes with electrolytes. The LCM activity coefficient equations are explicit functions, and require computational methods. Due to length and complexity, only the salient features of the LCM equations will be reviewed in this paper. The Aspen Plus Electrolyte Manual (1) and Taylor (21) present the final form of the LCM binary and multicomponent equations used in this work. 

Method Explicit Matrix Relations for Total Exchange Areas, Int.J. Heat Mass Transfer, 18, 261-269 (1975). Rhine, J. M., and R. J. Tucker, Modeling of Gas-Fired Furnaces and Boilers, British Gas Association with McGraw-Hill, 1991. Siegel, Robert, and John R. Howell, Thermal Radiative Heat Transfer, 4th ed., Taylor Francis, New York, 2001. Sparrow, E. M., and R. D. Cess, Radiation Heat Transfer, 3d ed., Taylor Francis, New York, 1988. Stultz, S. C., and J. B. Kitto, Steam Its Generation and Use, 40th ed., Babcock and Wilcox, Barkerton, Ohio, 1992. 

The alternating direction implicit (ADI) method (Peaceman and Rachford, 1955) is a partially implicit method. The equation is rearranged so that one coordinate may be solved implicitly using the Thomas algorithm whilst the others are treated explicitly. If this is done alternately, each coordinate has a share of the implicit iterations and the efficiency (Gavaghan and Rollett, 1990) as well as the stability is improved. The method was used by Heinze for microdisc simulations (Heinze, 1981 Heinze and Storzbach, 1986) and has subsequently been adopted by others (Taylor et al, 1990 Fisher et al., 1997). 

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit w— 0. /3 values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B ) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth phenomenological convention (X) is converted to the T convention by multiplication by a factor of 4. 

Fig 6.25 The hierarchy of cells used in the cell multipole method For an atom in the black cell, the interactions with atoms in the 26 nearby cells (N) are calculated explicitly Interactions with the atoms in cells labelled A and B are calculated using a Taylor series multipole expansion. (Figure adapted from Ding H-Q, N Karasaxva and W A Goddard III, 1992b The Reduced Cell Multipole Method for Coulomb Interactions in Periodic Systems with Million-Atom Unit Cells Chemical Physics Letters 196-6-10)... 

Frey and Rodrigues [20] proposed a method for the explicit calculation of multi-component equilibria using I AST. They used a polynomial or Taylor series to approximate the relationship between K and However, their method has... 

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