Control of step size

Beyond the choice of the type of method, the discretization error depends on step size for a method of given order. An automatic control of step size is needed, not only to avoid too many large errors, but also to minimize the number of steps. Let us assume that the local discretization error, e, is of the form 

From these two relationships, it is possible to compute the maximum step size, bmax, if the error e associated with a step size h may be evaluated, viz. 

Damping factors are generally used to avoid excessive variations of h. 

The evaluation of the local discretization error may be achieved by three kinds of method. 

The integration process is carried out, from f 2 to t , first in two steps of step size h, then in a single step of step size 2h. From the difference between the corresponding values xand x h an estimate of the local discretization error can be computed, viz. 

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A way to check the accuracy of the solution is repeating the procedure with a smaller step size until the significant digits will be unchanged. More efficient methods of step size control will be discussed in Section 5.1.3. In this example the step size h = 0.05 hours has been proved appropriate and results in the following solution ... 

To control the step size adaptively we need an estimate of the local truncation error. With the Runge - Kutta methods a good idea is to take each step twice, using formulas of different order, and judge the error from the deviation between the two predictions. Selecting the coefficients in (5.20) to give the same a j and d values in the two formulas at least for some of the internal function evaluations reduces the overhead in calculation. For example, 6 function evaluations are required with an appropriate pair of fourth-order and fifth-order formulas (ref. 5). 

In addition to the restrictions that stability places on the step size hn, we also need to be concerned with how accuracy affects the choice of step size. Assume that the local accuracy is to be controlled to within a certain tolerance e and that accuracy can be estimated by the local truncation error. The time step must be chosen to keep a norm of the local truncation error below the tolerance, that is ... 

Kohler, D., Ruff, C., Meyhofer, E., and Balder, M. (2003). Different degrees of lever arm rotation control myosin step size./. Cell Biol. 161, 237-241. 

An ideal solution to the problem of finding a universal method for making monodisperse imprinted beads would be a combination of the inertness of fluori-nated liquids with the quality of beads produced by the two-step swelling process. Unfortunately, to date it has not proved possible to make a microemulsion of the organic phase in a liquid fluorocarbon. It might be possible to get some control of particle size, however, by swelling a seed latex with the smallest droplet-size suspension which can be made. Work is in progress to evaluate this possibility. Other methods are available to achieve better particle size control in suspension polymerisation and these should also be evaluated [54]. 

Chemical, thermal, photochemical and structural stability one-step procedure at ambient or low temperature Control of pore size and morphology high electro chemical active surface area high conductivity suitable for development of miniaturized biosensor devices Suitable for miniaturization enhanced sensitivity low cost production (usually lithography)... 

The range of substances over which there is pharmacopoeial control of particle size is shown in Table 1.7 sometimes the aim is to achieve uniformity in a product rather than any direct benefit. The control exercised over the particle size of cortisone acetate and griseofulvin is due to their very low solubility the experience is that if the solubility of a dmg substance is about 0.3% or less then the dissolution rate in vivo may be the rate-controlling step in absorption. 

As noted earlier, when crystallization is employed as a crude fractionation step earlier in the purification train, larger particles are desired to facilitate centrifugation or filtration of the slurries. Control of particle size during these crystallizations, therefore, is of great importance due to its influence on subsequent handling steps. 

Many other patents on the production of detergents by agglomeration, granulation or two step processes have issued. Several advantages are often claimed for such processes such as control of particle size, better properties for the finished material, economy, etc. In most of these processes, the silicates play an important role due to their ability to act as a structure former-binder as well as for their effect on detergency. 

The improved two-step process permits control of particle size within a narrow angle. 

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