Bickley formulae

W. G. Bickley. Formulae for Numerical Differentiation. Mathematical Gazette, 25 (1941) 19-27. 

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]. 

Inverting the matrices and multiplying out the second row with the coefficient vector finally yields the approximation, presented in Table A.2 in Appendix A, together with a few others. It turns out that in the process, the terms in h5 drop out and the final approximation is of 0(/i4), arising from the neglected terms in h6. The formula has been given as early as 1935 by Collatz [169], who also presented some asymmetric forms in his 1960 book [170], and Bickley in 1941 [88]. Noye [423] also provides a number of multipoint second derivatives for use in the solution of pdes. 

The BDF method is ascribed to Curtiss k Hirschfelder [188], who described it in 1952, although Bickley [88] had essentially, albeit briefly, mentioned it already in 1941. Considering Fig. 4.3, the method can be seen as a multipoint extension of BI the derivative y is formed by using a number k of points from y. k i > to yn+1, but referred to the new point yn+. This implies a backward derivative, with formulas of the form y n(n) as in Appendix A, Table A.l. For example, using the three points shown in Fig. 4.3 (in other words, k = 3), the table yields the formula... 

Investigations of the perturbation equations for the rotating sphere have been carried out more or less independently by a number of individuals. The first of these appears to be due to Bickley (B5), who obtained a solution correct to the first order in R. His solution indicates an inflow of fluid towards the poles and a corresponding outflow around the equator. Rather interestingly, to this order in R the torque itself is unaffected and continues to be given by the Stokes formula... 

Simple discretization schemes use derivatives of Lagrangian interpolation polynomials that approximate the function known only at the grid points x,. These schemes consist of tabulated numbers multiplied by the function s values at m contiguous grid points and are referred to as "m-point-formulae" by Bickley [498] (cf. Ref. [468, p.914]). For an acceptable truncation error 0(h ), f = 4 or higher, m is larger than t which leads to an extended amount of computation. 

M-point Bickley central-difference formula with sufficiently high n may guarantee sufficient accuracy even in cases where relativistic effects on the gradient are negligible. 

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