Adams Methods

If j30 = 0, the method is explicit and the computation of is straightforward. If 30 + 0, the method is implicit because an implicit algebraic equation is to be solved. Usually, two algorithms, a first one explicit and called the predictor, and a second one implicit and called the corrector, are used simultaneously. The global method is called a predictor-corrector method as, for example, the classical fourth-order Adams method, viz. 

Predictor-Corrector methods have been constructed attempting to combine the best properties of the explicit and implicit methods. The multistep methods are using information at more than two points. The additional points are ones at which data has already been computed. In one view, Adams methods arise from underlying quadrature formulas that use data outside of specifically approximate solutions computed prior to t . 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called the Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to /(t TO, V )i. .., f tn, explicit method of order m- -1. Methods of this tirpe are called Adams-Bashforth methods. It is noted that only the lower order methods are used for the purpose of solving partial differential equations. The first order method coincides with the explicit Euler method, the second order method is defined by ... 

The reduction of nitrobenzene with cyclohexane, cyclohexene, and n-hexane over a series of catalysts on aluminum oxide afforded aniline in yields greater than 90%. An iridium catalyst prepared by Adams method is useful for the reduction of nitroaromatics to hydroxylamines... 

Method of Fabrication. The electrocatalyst, in the form of fine powder (15), can be produced by the Adams method (18). It is first mixed with PTFE emulsion solution such that the catalyst/PTFE ratio is 80 20 (19). The mixture is placed on a metal foil and sintered at 345 for more than one hour (20). After cooling, the Teflon-bonded catalyst is transferred from the foil to the membrane. The sintering process can be... 

J.A. Lawton and J.T. Adams, Method for fabricating an integral three-dimensional object from layers of a photoformable composition, US Patent 5122 441, assigned to E. I. Du Pont de Nemours and Company (Wilmington, DE), June 16,1992. 

These moment equations are typically integrated by linear multistep methods, such as the Adams method and the backward differentiation formula (Petzold, 1983). The right-hand sides of Eqs. (10.6), (10.7) are directly related to the selected reaction scheme. As an example. Scheme 10.1 shows the main reaction steps for a simplified chain-growth radical polymerization. 

ODE45 Runge-Kutta fourth-order method with fifth-order error prediction. ODE23s Rosenbrock. Implicit low-order method for stiff problems. ODE113 Adams method. Multistep method. 

According to the Bashforth-Adams method, the interfacial tension between a pendant drop of a liquid and its surrounding medium is related to the drop geometry as follows [25,26] ... 

The idea leading to Adams methods is quite simple. It is based on transforming the initial value problem... 

Example 4.1.5 The local residual of the two-step implicit Adams method is defined by... 

Thus the implicit two step Adams method has the order of consistency 3. 

As for Adams methods p x) = - x all Adams methods are zero stable. 

However, for the use in switching algorithms, see Sec. 6, the interpolation error is propagated as it influences the determination of the switching point and the point of restart. Thus in this case, the interpolation error must be controlled. In [SG84] it is shown for Adams methods that the continuous representation is error controlled, for BDF methods the corresponding result can be found in [EichQl]. 

It should be noted that the way an Adams method is implemented influences the continuous representation of the solution. Special care has to be taken in P EC) implementations, see Sec. 4.1.1. In that case X (l) a n+i- For this type of implementation (4.5.1) deflnes no continuous representation of the solution. Using (4.5.1) in that case may lead to problems when passing from one interval to the next in connection with switching algorithms, which will be described in Sec. 6. 

Also methods using past values of the right hand side like Adams methods can at least formally be applied to DAEs. We will start here with the trapezoidal rule and will treat a general multistep formulation in the next section. 

For BDF or Adams methods such a representation can be obtained directly from the construction idea of the discretization method using the polynomial already computed for the discretization. 

In [SG84] it is shown for Adams methods that this representation is error controlled, for BDF methods the corresponding result can be found in [Eich91]. 

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