Discretisation of boundary conditions

This subroutine is a general routine for computing the first or second derivative on a number n of points, referred to the ith one among that number, on an arbitrarily spaced grid of points. The derivatives are computed as a linear sum of terms, and the coefficients in that sum are also passed back, for use, for example, in the discretisation of boundary conditions or the spatial second derivative. The number of points is in principle unrestricted, but the routine will fail for values n > 12, where the accuracy abruptly drops. A value, in any case, exceeding about 8, is perhaps impractical. This routine can be used instead of the algebraic expressions shown in Chap. 7, or if n values greater than 4 are required. 

This seems a poor, low-order approximation. It can be justified, however, in cases where H is very small, as is in fact so with most useful programs these days, since these use unequal intervals, usually spaced very closely near the electrode. As wiU be seen, this two-point form makes the discretisation of boundary conditions much easier. There are even cases in which the current approximation becomes worse as more points are introduced. This happens with severely stretched grids (see unequal intervals , elsewhere), so the n-point formula should probably be used only with equal intervals. It has also been argued [9] that the three-point formula for equal intervals,... 

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