Three-point formulas

Y.P. Chiou, Y.C. Chiang, H.C. Chang, Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices , J. Lightwave Technol. 18, 243-251 (2000). 

To calculate y(x) at any point x, Eq. 11.9 uses three points Xo,x,X2 with X() < X < X2, and is called the Lagrange three-point interpolation formula. The three-point formula amounts to a parabolic representation of the function y(x) between any three points. 

The error associated with the three-point formula is (applying again Eq. 

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,... 

Firstly, the discretisation itself is described. We restrict the discussion to the BI time integration, in order to focus on the spatial discretisations. The program UMDE DIRECT in fact uses BI as the first step, then three-point BDF, which produces second-order accuracy with respect to 8T, this being the rational BDF startup described in Chap. 4, page 70. Take a point away from the boundaries, indices i (for Z) and j (for R). The discretisation at concentration Cij of the pde (12.17) has three derivative terms, all to be discretised using three-point formulas. The coefficients can be precalculated. For the row along Z, there are, for each 0 < Z < Z ax, that is, 0 < / < z, four coefficients for the approximations... 

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