Unequal intervals

In this chapter, only one-dimensional unequal intervals will be described. Mapping techniques for two-dimensional simulations are left to Chap. 12. 

Consider Fig. 2.4 on p.16, showing the concentration profile for a Cottrell simulation at different times. It is clear that especially the profiles at small T values are strongly compressed near the electrode, and that equal intervals in X would be wasteful at larger X. An unequal spacing of the intervals could not only provide more detail near the electrode where it is needed, but also make do with fewer points by wide spacing far away from the electrode. So some kind of grid stretching is indicated on this account. 

Dieter Britz Digital Simulation in Electrochemistry, Lect. Notes Phys. 000, 103—117 (2005) www.springerllDk.com c Springer-Verlag Berlin Heidelberg 2005 

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, 

Equation (3.25) holds for equal intervals but also for arbitrary (unequal) intervals, if the coefficients are computed accordingly. For unequal intervals, the coefficients must be computed (probably precomputed in a given program), as they cannot be tabulated, and this is best done using the Fornberg algorithm [10], to be described in the later Chap. 7. It is implemented in the routine GOFORN, also described in Appendix E. 

6 High-Order Compact (Hermitian) Current Approximation 

The information on derivatives that the device makes use of is the pde itself, which can be written in the form 

First we consider the current approximation presented in the above two sections. A question left untouched, for example, the equation for the current approximation (3.25) above, is just what terms were dropped when generating a particular form. The order of what was dropped is given in Sect. 3.3, but not extended to actual higher terms. This must be done now. Bieniasz [11] presents a table of these and we can write the first few. For this, it is convenient to use a more compact notation for the higher derivatives let 

In reality, this is not possible because we never have a stationary and known concentration profile (if so, why simulate it ). However, a spreading function can still help. If well chosen, it can not only improve accuracy near the electrode where it counts but substantially reduce the number of concentration points as well. Joslin and Fletcher (1974) found that, for the same accuracy as the standard box method, this technique can reduce computing time by up to a factor 100 - a worthwhile saving. Even a more modest factor of 10 would be nice. 

Just what constitutes a well chosen function is somewhat of an art. One wants a function that looks roughly like the concentration profile in Fig. 5.3 if (as is in fact sometimes the case) that profile is of the error function form, then the transformation 

Feldberg s actual function is an exponentially expanding sequence of box widths and has been used previously in non-electrochemical work (Pao and Daugherty 1969, cited by Noye, 1982). 

0183 value and X,. 6 would have implied about 330 samples, six times 1 im 

The actual value chosen for the parameter a, does not seem to have strong effects on the efficiency of the transforming function, Eq. 5.69. Trials suggest that a 2 or 3 is about optimum - at higher values, some instabilities make themselves felt and although they damp out again, they cause some loss of accuracy. 

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Of the related polysaccharides mentioned earlier (see p. 267), only arabinoxylans have been examined in any detail by enzymic methods. Goldschmid and Perlin153 studied the fine structure of wheat arabinoxylan by using the /3-D-xylanase from Streptomyces QMB 814. Their results indicated that arabinoxylan molecules are mainly constituted of highly branched regions in which isolated and paired L-arabinosyl (A) branches are separated by single D-xylosyl (X) residues, as shown in 12, but that, at unequal intervals (averaging... 

Wiesner s expression used different symbols, but this is not important.) This expression strictly holds only for a first-order reaction and Vetter [559] provides a more general expression. However, the above expression is sufficient for most simulation purposes. The equation for fi holds in practice only for rather large values of the rate constant for small values below unity, fi becomes greater than the diffusion layer thickness, which will then dominate the concentration profile. At the other end of the scale of rate constants, for very fast reactions, // can become very small. The largest rate constant possible is about 1C)1 ° s 1 (the diffusion limit) and this leads to a fx value only about 10 5 the thickness of the diffusion layer, so there must be some sample points very close to the electrode. This problem has been overcome only in recent years, first by using unequal intervals, then by the use of dynamic grids, both of which are discussed in Chap. 7. 

Formally, the above process is equivalent to (6.4), extended for any n and solving that system. The u-v device is a more efficient way of solving it than any linear equation solver that might otherwise have been used, as n becomes larger. The u-v device will be extensively used in this book, even with implicit methods for coupled equation systems, where we must solve for a number of concentration profiles (see below). There are practitioners who believe that n = 2, that is the two-point G-approximation, is good enough. This is justified in cases where H is very small, as it often is, at least near the electrode, when unequal intervals are used (see Chap. 9). In that case, one can simply use (6.5). 

Transformation for electrochemical work was proposed in the now classic paper by Joslin and Pletcher [321], They described a transformation, say from X to Y, such that equal intervals in Y are a mapping of (correspond to) unequal intervals in X. The aim is to find a transformation function which produces in T-spaee a concentration profile that resembles a straight line as much as possible. 

The Laasonen method, because of the forward difference in T, has errors of 0(6T, H2), and the first-order behaviour with respect to ST limits its accuracy to about the same as the explicit method described in Chap. 5. However, it has a smooth error response to disturbances such as an initial transient (Cottrell), and is stable for any value of 6T/H2, where // is either the same as all intervals if equal intervals are used in X, or is the smallest (usually the first) intervai if unequal intervals are used. This makes the method interesting, and it will be seen below that it can be improved. For simplicity, the symbol A will be used below, and denotes the largest value of that parameter, that is, the value from the smallest interval in space in a given system. 

The method works well if A 1 or, in the case of unequal intervals or two-dimensional geometries, where there is some critical, largest effective A greatly exceeding unity. It was found 1149] that the method works very well with a single BI step in the case of (2-D) microdisk simulations, where indeed large effective A values result at the disk edge and it is these that are responsible for the oscillations if CN is used. 

The problem of thin reaction layers are described sufficiently in Chap. 5. The solution is to use unequal intervals, that is, a few very small intervals near the electrode, so that there are sample points within the thin profile. This can be done up to a point by a fixed unequal grid such as the exponentially expanding grid described in Chap. 7. A more flexible approach is the moving adaptive grid also described in that chapter. This problem is thus solved and needs no further attention here. 

In practice, the (6,5) approach is, at present, limited by the fact that, in the form presented here, it applies to equal intervals. A slight improvement with unequal intervals, using a 4-point spatial second derivative, is described in Chap. 8, and this might be sufficient improvement, at little cost in terms of desk work [143]. It has been applied to the ultramicroelectrode [532], see Chap. 12. 

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). 

It was soon realised that at least unequal intervals, crowded closely around the UMDE edge, might help with accuracy, and Heinze was the first to use these in 1986 [300], as well as Bard and coworkers [71] in the same year. Taylor followed in 1990 [545]. Real Crank-Nicolson was used in 1996 [138], in a brute force manner, meaning that the linear system was simply solved by LU decomposition, ignoring the sparse nature of the system. More on this below. The ultimate unequal intervals technique is adaptive FEM, and this too has been tried, beginning with Nann [407] and Nann and Heinze [408,409], and followed more recently by a series of papers by Harriman et al. [287,288,289, 290,291,292,293], some of which studies concern microband electrodes and recessed UMDEs. One might think that FEM would make possible the use of very few sample points in the simulation space however, as an example, Harriman et al. [292] used up to about 2000 nodes in their work. This is similar to the number of points one needs to use with conformal mapping and multi-point approximations in finite difference methods, for similar accuracy. 

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. 

In Chap. 7, two ways of implementing unequal intervals were described. These were the Feldberg approach, in which exponentially expanding boxes are placed along the X-axis (7.16), and the transformation method (7.3). Here it will be shown that they are approximately equivalent, and the relation between their respective expansion parameters will be given. 

Note that the use of both GU and CU is not restricted to unequal intervals they can also be used with equal intervals, where we already have GOFUNC and COFUNC, given above. The present two functions will take a little more computing time, but this is normally a small part of any given simulation, where the recalculation of a concentration profile is most time consuming. 

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 program is again a Cottrell simulation using second-order extrapolation based on the Bl (Laasonen) method and unequal intervals, but in contrast with the above program C0TT EXTRAP, this one makes use of the four-point spatial derivative approximation, and the GU-function. It performs a little better than the above program, at little extra programming effort. 

The program LSV4IRC is a simulation of a reversible reaction with input values of p (dimensionless uncompensated resistance and qc (dimensionless double layer capacity). Unequal intervals are used, with asymmetric 4-point second spatial derivatives, and second order extrapolation in the time direction. The nonlinear set of 6 equations for the boundary values is solved by Newton-Raphson iteration. Some results are seen in Chap. 11. 

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