Kimble and White KW

In the literature, the schemes are usually described not in terms of fractional steps but with a number of whole-interval steps the two descriptions are equivalent, however, and it seems that a combination of fractional steps, ending with a new value at the next time interval, is more convenient. 

The method due to Kimble and White [17] is not actually a method designed for ode%, but was devised by the authors for electrochemical pde%. The method can however be easily adapted to odes and in fact might be more appropriate there. The method described in 1990 had a precursor in 1987 [26] and this section will start with a description of its expression for odes, because it is simpler and makes the point more clearly. A cut-down application of it has already been outlined in Sect. 4.8.1. 

The essence of KW is that multi-point central differences are used as derivatives along most of the t scale, with some asymmetric expressions necessarily added at the ends. Rather than using the time-marching method that is common to all the methods described in previous sections, KW puts all the approximations into one large system of equations, and solves the lot. It turns out that this results in a formitous stability [27]. 

The method is based on another time-marching scheme not mentioned in the above sections the leapfrog method [28, 29], also called the midpoint rule by Hairer and Wanner [6], using central differences. Equation (4.1) can be approximated as 

The interesting thing is that all but the last equation are leap-frog forms, which by themselves result in an unstable solution if using them in a time-marching manner. The mere addition of the last equation (4.39) renders the system stable, and the solution is of 0(8f). 

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Kimble and White [14] developed a scheme which, as described and intended, was somewhat awkward to use and limited the possible number of points in time and space. The method is mentioned in other chapters for its use as a high-order start for BDF (for which it did indeed work, but not with great efficiency). It is perhaps best described in two stages. Consider Fig. 9.2, a modest-sized grid on which the KW method is to be used, representing positions in time (indices j) and space (indices i). The thicker bottom line represents initial conditions the dotted line at the left is that... 

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