Approximations multipoint derivatives

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

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. 

Higher-order methods Chap. 9, Sect. 9.2.2 for multipoint discretisations. The four-point variant with unequal intervals is probably optimal the system can be solved using an extended Thomas algorithm without difficulty. Numerov methods (Sect. 9.2.7) can achieve higher orders with only three-point approximations to the spatial second derivative. They are not trivial to program. 

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