Multi-Point First Derivative Approximations

Table A.l. rnfi (3.14) for multi-point first derivatives. The notation y[(n) means the approximation at point, i using n points nmnftered 1. .. n...

The above approximations to a first derivative used only two points, which sets a limit on the approximation order. By using more points, higher-order approximations can be achieved. In the context of this book, forward and backward multi-point formulae are of special interest, as well as some asymmetric and centra] multi-point ones. To this end, a notation will be defined here. Figure 3.2 shows the same curve as Fig. 3.1 but now seven points are marked on it. The notation to be used is as follows. If a derivative is approximated using the n values yi. . yn, lying at the x-values. iq. ..xn (intervals h) and applied at the point (Xi,yi), then it will be denoted as y (n) (for a first derivative) and y"(n) (for a second derivative). 

The above approximations to a first derivative used only two points, which sets a limit on the approximation order. By using more points, higher-order approximations can be achieved. In the context of this book, forward and backward multi-point formulae are of special interest, as well as some asymmetric and central multi-point ones. To this end, a notation will be established here. Figure 3.2 shows the same... 

Spatial discretisations on unequally spaced points are best done using multi-point stencils, for example five-point. However, in order to keep things simple, three-point approximations are used in what follows. The symbols like i)k refer to three fi coefficients k = 1,2,3 pertaining to point i, used to approximate a first derivative at point i, and similarly for the a coefficients. These are all precomputed using the Fornberg algorithm [63], implemented in the subroutines forn and fornberg, described in Appendix E. 

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