Local extrapolation

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. 

It is known that the local error estimate is obtained to the lower-order solution. This is applied, also, in our case of the local error estimate, i.e. the local error estimate is obtained for j>J+1. However, if this error estimate is acceptable, i.e. less than the bound acc, we consider the widely used local extrapolation technique. Thus, although we are controlling an estimation of the local error in the lower-order solution y%+l, we use the higher-order solution y f+l at each accepted step. 

In this case only the error of the lower order method can be estimated. Nevertheless, this quantity is frequently used to control the step size of the higher order method (2). This technique is called local extrapolation. 

Though it is always the error of the lower order method which is estimated, one often uses the higher order and more accurate method for continuing the integration process. This foregoing is called local extrapolation. This is also reflected in naming the method, i.e. 

This method uses six stages for 5 order result and one more for obtaining the 4 order result One clearly sees that the method saves one function evaluation by meeting the requirement (4-2.5) for the 5th order method which is used for local extrapolation. 

However, before extrapolating the arguments from the gross patterns through the reactor for homogeneous reactions to solid-catalyzed reactions, it must be recognized that in catalytic reactions the fluid in the interior of catalyst pellets may diSer from the main body of fluid. The local inhomogeneities caused by lowered reactant concentration within the catalyst pellets result in a product distribution different from that which would otherwise be observed. 

For large systems comprising 36,000 atoms FAMUSAMM performs four times faster than SAMM and as fast as a cut-off scheme with a 10 A cut-off distance while completely avoiding truncation artifacts. Here, the speed-up with respect to SAMM is essentially achieved by the multiple-time-step extrapolation of local Taylor expansions in the outer distance classes. For this system FAMUSAMM executes by a factor of 60 faster than explicit evaluation of the Coulomb sum. The subsequent Section describes, as a sample application of FAMUSAMM, the study of a ligand-receptor unbinding process. 

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

Depth of localized corrosion should be reported for the actual test period and not interpolated or extrapolated to an annual rate. The rate of initiation or propagation of pits is seldom uniform. The size, shape, and distribution or pits should oe noted. A distinction should be made between those occurring underneath the supporting devices (concentration cells) and those on the surfaces that were freely exposed to the test solution. An excellent discussion of pitting corrosion has been pubhshed [Corro.sion, 25t (January 1950)]. 

As with all elec trochemical studies, the environment must be electrically conduc tive. The corrosion rate is direc tly dependent on the Tafel slope. The Tafel slope varies quite widely with the particular corroding system and generally with the metal under test. As with the Tafel extrapolation technique, the Tafel slope generally used is an assumed, more or less average value. Again, as with the Tafel technique, the method is not sensitive to local corrosion. 

These figures can be used for predictive purposes to extrapolate average major incident conditions to situations under study, provided the actual conditions under study correspond reasonably well with average major incident conditions. Such a condition may be broadly described as a spill of some tens of tons of a hydrocarbon in an environment with local concentrations of obstructions and/or partial confinement, for example, the site of an average refinery or chemical plant with dense process equipment or the site of a railroad marshaling yard with a large number of closely parked rail cars. It must be emphasized that the TNT equivalencies listed above should not be used in situations in which average major incident conditions do not apply. 

Interim reports document the installation process and provide a way to flag opportunities and problems to management s attention in a timely manner, which helps avoid surprises. At the same time, be careful that your interim reports are precisely that don t try to second-guess final reports, or extrapolate from local findings prematurely. 

Although fluxes of precipitation and river discharge can be quite accurately determined on a local scale, large portions of the globe, especially the oceans and Antarctica, are essentially ungauged, requiring extensive extrapolation of existing data. Evaporation fluxes are even less well known, since calculation requires... 

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