Kimble White method

In this way, the coefficients for any y((n) can be calculated. Table A.l in Appendix A shows them all, as whole numbers m/3j, where m is the multiplier mentioned above. For each n, the Table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. For n up to 6, all possible forms are included, as they will be needed later, while for n = 7, only the forward and backward formulae are shown, as only these are needed. In case the reader wonders why all this is of interest the forms y[(n) will be used to approximate the current in simulations (see the next section) the backward forms y n(n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the Table will be used for the Kimble White (high-order) start of the BDF method, also described in these chapters. The coefficients have a long history. Collatz [169] derived some of them in 1935 and presents more of them in [170]. Bickley tabulated a number of them in 1941 [88], The three-point current approximation, essentially y((3) in the present notation, was first used in electrochemistry by Randles [460] (preempted by two years by Eyres et al. [225] for heat flow simulations), then by Heinze et al. [301], and schemes of up to seven-point were provided in [133]. 

This, as has been mentioned above, was not the way Kimble White applied the method, but it works rather well and may well be more practical for odes. In their application to pdes, the authors reduced the field to only a few points in both time and space, because otherwise the matrix to be solved, even when restricted to the pentadiagonal form, becomes very large. 

The method due to Kimble Sz White [338] is not actually a method designed for odes, but was devised by the authors for electrochemical pdes. 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 [414] 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. 

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... 

This early paper was followed by another one in 1990 by Kimble and White [338], 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. 

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 5 points for all approximations but their table of discretisations shows that they used 6 points at the edges for the spatial second derivative. This is no doubt because, as Collatz already mentions in 1960 [170], the asymmetric 5-point second derivative is only third-order, while a 6-point formula is fourth-order, like the symmetrical 5-point ones used in the bulk of the grid. So, for the second spatial derivative at index i = 1, the form 2/2(6) was used, and the reverse, form 2/5 (6) at i = N. 

Wu and White [577] have described a new method that is reminiscent of the earher work of Kimble and White [338] but makes use of the Hermitian method (that is, using derivatives) to achieve higher-order solutions for several concentration rows at a time. They also suggest, but do not demonstrate, the use of their new scheme as a possible start-up for BDF. The reader is referred to their paper for details. 

Kimble MC, White RE (1990) A five-point finite difference method for solving parabolic differential equations. Comput Chem Eng 14 921-924... 

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... 

Wu and White [98] have described a new method that is reminiscent of the earlier work of Kimble and White [14] but makes use of the Hermitian method (that is. 

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