Multistep method

However, we could also consider a multistep formulation  

We can approximate the function f x,t) by an interpolating polynomial passing through points tj, fj). This interpolating polynomial will be called and is given by 

There are two types of interpolating polynomials that can be used. These are open formulas which are used to predict the Xj+i value based on known information up to Xj, and closed formulas which are used to correct the Xi+i. In both cases backward difference interpolating polynomials are used since we are using previous time information to determine current or future time behavior. In order to develop the backward difference formulas, we use Newton s fundamental formula for interpolating polynomials, 

In order to generate (pnit) using backward differencing, we have 

In order to avoid the sign change problem associated wdth equation (3.3.6), we let 

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. 

Euler s method can be considered as the simplest form of the Adams-Bashforth method, where the function / is represented by a polynomial p of order 1. If p is assumed to be a linear function that interpolates between (xfe i, /fc-i) and (xfc, fk), then p can be represented by  

Similarly, one can obtain higher-order Adams-Bashforth methods by using higher-order polynomials. 

In the improved Euler method (5.14) we use derivative information at two points of the time interval of interest, thereby increasing the order of the method. A straightforward extension of this idea is to use the derivative at several grid points, leading to the -step formulas 

The multistep method (5.25) is explicit if bQ = 0, otherwise it is implicit. These latter are the best ones due to their improved stability properties. To use an implicit formula, however, we need an initial estimate of yi+1. The basic idea of the predictor - corrector methods is to estimate y1+1 by a p-th order explicit formula, called predictor, and then to refine yi+1 by a p-th order implicit formula, which is said to be the corrector. 

Repeating the correction means solving the algebraic equation (5.25) by successive substitution. The use of more than two iterations is not efficient. 

The great advantage of the predictor - corrector methods is that in addition to y1+, in expression (5.25) we need only previously computed (and saved) function values. Thus, the computational cost depends on the number of corrections and does not depend on the order p of the particular formula. 

Starting a multistep method is an additional problem, since no previous function values are yet available. One can start with a one step formula and a small step size, then gradually increase to the desired value. A more common approach is to use Runge - Kutta steps of the same order at the beginning. 

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This one-step procedure is a convenient and general method for the preparation of carbamates. It is substantially simpler, quicker, and safer than the multistep methods hitherto used for the preparation of carbamates of tertiary alcohols. This procedure is applicable to the preparation of carbamates of primary, secondary, and tertiary alcohols and mercaptans, polyhydric alcohols, acetylenic alcohols, phenols, and oximes. It has also been extended to the preparation of carbamyl derivatives (i.e., ureas) of inert (non-basic) amines.10... 

For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

Fabrication of catalyst immobilized microchannel reactors usually needs expensive, complex, and multistep methods. On the eontrary, reactors with simple structures (Figure 5) can also perform effieient hydrogenation reaetions. Yoswathananont et al. [21] reported an effieient hydrogenation reaction in a continuous flow system by the use of a gas-liquid-solid tube reactor. 

The synthesis of cyclic carbonates, starting from olefins, can be also carried out via a multistep method based on two separate reactions. To this end, C02 and the carboxylation catalyst have been added to the same reactor in which a preliminary epoxidation process had been carried out. 

Similarly, Ono et al. [189] have reported using a composed catalytic system, namely MTO/UHP/Zn[EMIm]2 Br4/[BMfm]BF4 (UHP, urea hydrogen peroxide and MTO, methyltrioxorhenium). With the multistep method described above, a... 

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)...

For this and other caveats regarding the multistep methods see Cramer CJ (2004) Essentials of computational chemistry, 2nd edn. Wiley, Chichester, UK, pp 241-244... 

Although there is no stable free S analog of CO, nevertheless, several CS compounds of iridium have been synthesized by multistep methods and their reaction chemistry has been investigated. " ... 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

Reiger and Ballschmiter [43] described a multistep method for PCA analysis in sewage sludge by cyclohexane-isopropanol extraction and cleanup on silica gel column chromatography. Fractionation on silica gel was achieved by eluting with hexane (FI), which desorbed hexachlorobenzene, 4,4 -DDE, PCB, PCDD, and PCDF. PCAs were then desorbed from the column with (90 10) hexane/di-ethyl ether. The recovery of PCAs by this method was 86%. These authors also noted that cleanup chromatography on activated alumina should be avoided because PCAs were either totally or partially destroyed by dehydrochlorination during the adsorption process. 

We will begin with the most direct method of alkylation, and then (in Sections 23.9 and 23.10) examine two older, multistep methods that are still used today. Direct alkylation is carried out by a two-step process ... 

The initial-value problem is solved using new, highly accurate formulas of the linear multistep method. 

In ref 152 the author produces methods based on numerical differentiation some classes of special multistep methods. For these methods the regions of absolute stability are shown. Numerical efficiency of the methods is examined by application of some of Henrici and of some methods obtained in this paper of the same order to a second-order initial value problem. 

G. Psihoyios and T. E. Simos, Trigonometrically-fitted symmetric multistep methods for the approximate solution of orbital problems. New Astronomy, 2003, 8(7), 679-690. 

F. Mazzia, A. Sestini and D. Trigiante, BS linear multistep methods on non-uniform meshes, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(1), 131-144. 

Gerald D. Quinlan and Scott Tremaine, Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits, The Astronomical Journal, 1990, 100(5), 1694-1700. 

A. D. Raptis, Exponential multistep methods for ordinary differential equations. Bull. 

J. Vigo-Aguiar and T. E. Simos, Family of twelve steps exponential fittingsymmetric multistep methods for the numerical solution of the Schrodinger equation, J. Math. Chem., 2002, 32(3), 257-270. 

J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial values problems, J. Inst. Math. Appl., 1976, 18, 189-202. 

D. S. Vlachos and T. E. Simos, Partitioned Linear Multistep Method for Long Term Integration of the N-Body Problem, Appl. Num. Anal. Comp. Math., 2004,1(2), 540-546. 

Zhongcheng Wang, P-stable linear symmetric multistep methods for periodic initial-value problems, Computer Physics Communications, 2005, 171, 162-174. 

P. S. Rama Chandra Rao, Special multistep methods based on numerical differentiation for solving the initial value problem. Applied Mathematics and Computation, 2006, 181, 500-510. 

Fig. 4. Multistep methods for eliciting specific antibody-producing cells (hybridomas). HAT, hypoxanthine, aminopterin, and thymidine PEG, polyethylene glycol.

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