Using Four-Point Spatial Second Derivatives

4 Using Four-Point Spatial Second Derivatives 

It was shown in Chap. 7 that the three-point second spatial derivative on an unequally spaced grid, leading to (8.1) with the coefficients defined in (8.3), can be improved with relatively small effort to an asymmetric four-point, formula, spanning the indices i — 1, i, % f 1, % I 2, with the second derivative referred to the point at index i. The diffusion equation is then semi-discretised to 

Note that this differs from the three-point system (8.11) in that every line now has four coefficients, and their definitions have been written as i-dependent, which is the case for exponentially expanding X positions. Note also that an extra point, index TV + 2, has been added to the row of X values. Point TV is still the last one to undergo diffusional changes, so the fact that the point TV + 2 lies rather far past Xu,n (which is at index TV), does not matter, both these extra points being either constant or set values, not subject to diffusional changes. 

The above system, although leading to a quadradiagonal system of equations, can still be solved by a smallish extension of the Thomas algorithm [153]. Consider the last two equations of (8.33) and rewrite them, putting the bulk concentration terms on the right-hand side  

We rewrite the last equation, which is already down to two unknowns, in the form 

In Appendix E, a few examples of the use of CN are described for a Cottrell simulation (COTT CN), chronopotentiometry (CHRONO CN, CHRONO CN HERM) andLSV(LSV CN). 

The extra two intervals Xv+i — X and X +i n+i need not be expanding and are best set equal to Xn — Xn-u Alternatively, one might choose asymmetric backward-pointing difference approximation m (4) for the point X -i and make Xn the bulk point, thus obviating the need for the extra two points. However, for methods where there is a Neumann (derivative) condition at the outer boundary, there is no such choice, and then Xn is the furthest point, and backwards derivative approximations must be applied on the boundary. This happens in such cases as thin layer eells, the diffusion domain discussed in connection with arrays of electrodes in Chap. 12, and a case the present authors have worked on, a flat polymer film containing an enzyme over a disk electrode [8]. 

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Firstly, the discretisation itself is described. We restrict the discussion to the BI time integration, in order to focus on the spatial discretisations. The program UMDE DIRECT in fact uses BI as the first step, then three-point BDF, which produces second-order accuracy with respect to ST, this being the rational BDF startup described in Chap. 4, page 59. Take a point away from the boundaries, indices i (for Z) and j (for R). The discretisation at concentration (r j of the pde (12.17) has three derivative terms, all to be discretised using four-point formulas. The coefficients can be precalculated. For the row along Z, there are, for each 0 < Z < Zmox, that is, 0 < i < n-z, four coefficients for the approximations... 

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 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. 

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. 

The above has considered arbitrarily spaced grids, whereas in practice, the spacing is often that of Feldberg [231]. In terms of points, the special case is a sequence of positions given by an exponentially expanding series of spatial intervals. This will be detailed in Chap. 7. Here, it is sufficient to mention that this special case makes the derivation of the coefficients for various derivative approximations easier and the expressions themselves more compact, as was reported by Martfnez-Ortiz [385]. That author also found that there is a particular value for the expansion parameter, q = /2j for which the asymmetric four-point second derivative, referred to the second of the four points u"(2,4), is third-order accurate, rather than second-order as for other parameter values or arbitrary placement of points. This can be of use in simulation. The four-point approximation has some good properties besides this, as will be explained in Chap. 7. 

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

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