Extended Thomas algorithm

An interesting special case, mentioned in Chap. 3, is that of the second derivative on four points, u"(4). For arbitrarily (unequally) spaced points, this is a second-order accurate approximation and, as described in Chap. 9, it has some advantages. It allows the use of an efficient extended Thomas algorithm, rather than a pentadiagonal solver or a sparse solver required if... 

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. 

The equation systems commonly found in the simulation of electrochemical problems have a diagonal structure with a bandwidth 2n+l such that a j = 0 for j > i + n and for i > j n. By particularising the LU decomposition for diagonal systems [extended Thomas algorithm) the above expressions turn into... 

Subsequently the extended Thomas algorithm discussed in Section 5.3 can be employed to proceed with the iterative Newton-Raphson method and calculate the concentration profiles and Ta/b values. Once these are known, the current response is given by the contributions of both electron transfer processes ... 

Unequal intervals Chap. 7. These are essential for most programs. The second spatial derivative requires four points if second-order is wanted (and is recommended). With four-point discretisation, an efficient extended Thomas algorithm can be used, obviating the need for a sparse solver. Very few points can then be used across the concentration profile. For two-dimensional simulations, direct three-point discretisation on the unequally spaced grid was shown to be comparable with using transformation and discretisation in transformed space. 

This is a formulation assuming equal intervals h in both directions so that we have A = 8T/h. It can easily be extended to unequal intervals. In practice, at each solution along a row or column, a simple tridiagonal equation system is solved, using the Thomas algorithm as already described in Chap. 8. 

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