The Kimble and White Method

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 

This early paper was followed by another one in 1990 by Kimble and White [14], now applying the method to a diffusion problem, and using 5-point approximations in both directions. As before, the problem was cast into a block matrix, but because of the 5 points used for the discretisations, this was block-pentadiagonal. For most node points in the figure, the 5-point approximations yield the following computational molecule or stencil. 

Similarly, asymmetric forward forms are used at the bottom end. 

Kimble and White were aware that leapfrog methods are unstable and simply remark that this did not seem to apply to their method. Also, they mention the use of five points for all approximations but their table of discretisations shows that they used six points at the edges for the spatial second derivative. This is no doubt because, as Collatz already mentions in 1960 [20], the asymmetric five-point second derivative is only third-order, while a six-point formula is fourth-order, like the symmetrical five-point ones used in the bulk of the grid. So, for the second spatial derivative at index i = 1, the form 3 2(6) was used, and the reverse, form yj(6) at i = N. 

Kimble and White [338] 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 

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