Unequal intervals discretisation

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. 

This is an example of a Cottrell simulation using second-order extrapolation based on the BI (Laasonen) method and unequal intervals. Three-point spatial discretisation is used here. 

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,... 

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. 

Just as space can be divided into unequally spaced intervals, so might time also be unevenly divided. As with spatial intervals, there is the choice between discretising on an uneven time grid or using a transformation to a new time scale. Since, except for BDF methods, one usually differentiates with respect to time using only two time points (levels), transformation does not make sense here. 

For unequal time intervals, we look at the example transformed diffusion equation 5.85. This is to be discretised at intervals of SO and H. We use index j to count 0 intervals and i to count X intervals (of H). 

A stretched stack of boxes was used by Seeber and Stefani and by Feldberg [7, 8] for the box-method, to be described in Chap. 9. Pao and Dougherty [17] developed the same idea (and stretching function) in 1969, in the context of fluid dynamic simulations. This is the simple placement of points at increasing intervals, in some suitable point distribution or stretching function, and discretisation of the second derivative of concentration along X on that unequal grid. 

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