Multistep method accuracy

CBS-Q, a high-accuracy multistep method with correlation energy correction and large basis sets (Section 5.5.2.2b)... 

We will concentrate on Gaussian-type and CBS methods, because these have been the most widely-used and have thus accumulated an archive of results, are the most accessible, and because several versions of them are available. However, there are other high-accuracy multistep methods, such as the Weizmann procedures of Martin and de Oliveira, W1 and W2 [188], and of Boese et al., W3 and W4 [189], which like the CBS methods are based on basis set extrapolation. W1 and W2 have a mean absolute deviation of about 1 kJ mol 1 (not 1 kcal mol-1), and incorporate relativistic effects, and W2 has no empirical parameters, unlike the Gaussian and CBS methods. W3 and W4 methods have similar errors to W1 and W2, and the authors speculate on the reasons for the obstinate 0.1 kcal/mol barrier . These very accurate methods are still limited molecules of about or less than the reach of CBS-APNO. 

Table 5.10 Comparison of speed and ability to handle molecular size for four popular high-accuracy multistep methods G3(MP2), CBS-4M, CBS-QB3, and CBS-APNO Time (h) for less than 1 h h (min)...

Note that these calculations of the heat of formation of methanol are not purely ab initio (quite apart from the empirical correction terms in the multistep high-accuracy methods), since they required experimental values of either the heat of atomization of graphite (atomization and formation methods) or the heat of formation of methane (formation method). The inclusion of experimental values makes the calculation of heat of formation with the aid of ab initio methods a semiempiri-cal procedure (do not confuse the term as used here with semiempirical programs... 

The multistep methods are quite cheap as they require only one evaluation of the derivative per time step. However, these methods require data from many points prior to the current one, thus they cannot be started using only data at the initial point. One has to use other methods to get the calculation started. A common approach is to use a small step size and a lower order method to achieve the desired accuracy, and slowly increase the order as more data becomes available. Moreover, the multistep methods often give accurate solutions for some time and then begin to behave badly as they can produce non-physical solutions that may grow. A useful remedy for this problem is to restart the method at certain intervals. However, this trick may reduce the accuracy and/or the efficiency of the scheme. 

This method requires that the positions (and forces) be known at two successive points h apart in time in order to initialize the iteration. These might be generated by using the Verlet method or some other self-starting scheme. Beeman s algorithm is explicit since, given q , q - andp , one directly obtains q + and then, q i, and thus p +i, with only one new force evaluation. Because it is a partitioned multistep method, its analysis is more involved than for the one-step methods, and, in particular its qualitative features are difficult to relate to those of the flow map. The order of accuracy of the scheme above can be shown to be three. 

Truncation error results from using a function like a polynomial to approximate the true solution. To reduce tmncation error, we could use more sophisticated functions—for example, higher order polynomials—which require more information. Multistep methods do this, but the additional accuracy gained by increasing the order of the method drops off very quickly beyond about k = 4. 

The implicit multistep methods add stability but require more computation to evaluate the implicit part. In addition, the error coefficient of the Adams-Moulton method of order k is smaller than that of the Adams Bashforth method of the same order. As a consequence, the implicit methods should give improved accuracy. In fact, the error coefficient for the imphcit fourth-order Adams Moulton method is 19/720, and for the explicit fourth-order Adams Bashforth method it is 251/720. The difference is thus about an order of magnitude. Pairs of exphcit and implicit multistep methods of the same order are therefore often used as predictor-corrector pairs. In this case, the explicit method is used to calculate the solution,, at v +i. Furthermore, the imphcit method (corrector) uses y + to calculate /(x +i,y +i), which replaces /(x +i,y +i). This allows the solution, y +i, to be improved using the implicit method. The combination of the Adams Bashforth and the Adams Moulton methods as predictor orrector pairs is implemented in some ODE solvers. The Matlab odel 13 solver is an example of a variable-order Adams Bashforth Moulton multistep solver. 

In one-step methods, the integration formula depends on previous steps (i.e., A - - 1-th point is calculated from fc-th point). However, since all these methods are dependent on polynomial approximations, the information about other points such as A — 1, A — 2, etc., is also included in the approximation which makes these methods more realistic. The multistep methods are dependent on more steps than just the previous one. The multistep methods are attractive because they provide better representation of the functional space and, hence, better accuracy. The following equation provides the generalized representation of the multistep methods where the function / is replaced by a polynomial function p. 

However, multistep methods suffer from two problems not encountered in one-step methods. One problem is associated with the starting of these methods. The k + 1-th step is dependent on k and A — 1 steps, etc., and at A = 1 there is no information about these previous steps. To circumvent this problem, a one-step method is used as a starter for a multistep method, until all the information is gathered. Alternatively, one may use a one-step method at the first step, a two-step method at the second, and so on, until starting values have been built up. However, it is important that the starting method used in subsequent stages maintain the same accuracy in the initial stages as the multistep method (which means that, initially one must use a smaller step size). 

The second problem with multistep methods is the presence of extraneous solutions. This comes under the category of techniques where the order of the solution method is more than the order of differential equations (m > q). Therefore while these methods provide greater accuracy, they may also possess strong instability characteristics. The following paragraphs describe some of the multistep methods. 

Term (2) calculated by the high-accuracy multistep CBS-APNO method (Section 5.5.2.2b) was 341.2 kcal mol-1 or 1426 kJ mol-1. The Sackur-Tetrode equation for the gas-phase entropy of the proton was mentioned in this regard, but in fact the algorithm automatically handles this. 

To improve the accuracy of the method an implicit multistep scheme is taken into consideration ... 

Defining characteristics of apparatuses for clearing of gas emissions are efficiency of separation and water resistance. Known methods of calculation of efficiency of the scrubbers based on the use of empirical functions, presenting parameters of fractional efficiency and dispersion composition of a dust, do not vary the significant accuracy. It is caused by the functions of mass particle size distribution of many industrial dusts which do not answer a logarithmically normal [lognormal] distribution because of the actions of several mechanisms of a dust formation. Use of some methods are inconvenient because of multistep, complexities, and labor contents of reception of initial data for calculation. 

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