Unequal intervals current approximation

There are simulation cases (for example using unequal intervals) where it is desirable to use a two-point approximation for G, both for the evaluation of a current, and as part of the boundary conditions. In that case, an improvement over the normally first-order two-point approximation is welcomed, and Hermitian formulae can achieve this. Two cases of such schemes are now described that of controlled current and that of an irreversible reaction, as described in Chap. 6, Sect. 6.2.2, using the single-species case treated in that section, for simplicity. The reader will be able to extend the treatment to more species and other cases, perhaps with the help of Bieniasz seminal work on this subject [108]. Both the 2(2) and 2(3) forms are given. It is assumed that we have arrived at the reduced didiagonal system (6.3) and have done the u-v calculation (here, only v and iq are needed). 

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,... 

There are the usual boundary conditions depending on the experiment performed on this system. One possible way to handle all this is simply to write out the whole system as a large linear system, expand that to include the boundary conditions, and solve. This, brute force approach (see below), has in fact been used [10, 11] and can even be reasonably efficient if the number of equations is kept low, by use, for example, of unequal intervals, described in Chap. 7. If the equations in such a system are arranged in the order as above (6.55), it will be found that it is tightly banded, except for the first two rows for the boundary conditions, which may have a number of entries up to the number n used for the current approximation. 

Feldberg and Rudolph [7] handle this by always choosing a very unequal spacing (in effect, large / ) that hopefully will be able to cope with the highest possible reaction rates. This means an extremely small first interval near the electrode, and then also justifies these authors use of the simple two-point approximation for the current (G). 

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