Direct Discretisation

One has the choice between applying finite difference discretisation either directly to a grid of points in the cylindrical (R, Z) space, or to a transformed space. In one dimension, it has been found [478] that direct discretisation without transformation is better. In the case of 2D simulations where edge effects are seen, this is not the case, and transformation is better. Both approaches are described here. 

There is a slight complication in the setting of the maximum R and Z values. The procedure depends on whether (12.17) or (12.26) is simulated. In 

If the characteristic time is defined independently of the disk radius, and diffusion (12.26) results, the Nernst diffusion layer thickness is dependent only on the number of these time units. So if the characteristic time is r and the maximum duration of the experiment is Tmax (giving Tmax = Tmax/r), then the final diffusion layer thickness is 1JDrmax. Then, in dimensionless distance units (normalisation being division by the disk radius a), this becomes, after multiplying by 6 and noting (12.27), 

This will become a little more complicated later, when the space is mapped into new coordinates, and limits in terms of these must be set. In Fig. 12.3, the number of nodes (lines) is held small, in order not to confuse the picture. The grid is chosen such that there are expanding intervals in both the Z direction and in the two R directions away from and on either side of the line R = 1. For reasons which will become clear below, the index j for the R positions starts at —1. There are n.i intervals between R = 0 and the disk edge, and a total of ur between the origin and the point at which R = Rmax, there are two further points beyond this, which also will be explained. The positions for Z, indexed with i, begin at zero and n-z is the point at which Z = Zmax. Again there are two further points beyond this value. 

We are now ready to apply the discretisations, but must decide on the vector of unknown concentrations at all the grid points in Fig. 12.3. It is convenient to include even the boundary points (but not those at j = —1, which serve only as fictitious points), setting these to known values in the large linear system to be generated. Thus we note that the total number N of unknowns is given by 

If the characteristic time is defined independently of the disk radius (as it is with LSV) and diffusion equation (12.27) results, the Nernst diffusion layer thickness is 

Note that P = 1 for the simulation of a potential step experiment, so that Eqs. (12.39) and (12.40) become identical to Eqs. (12.37) and (12.38). In the case of LSV Tmax is the time to scan a potential from a starting value, Estart, taking some potential units RT/nT, to the final potential Estop with a scan rate v. The characteristic time x was previously defined as the time to sweep through one p-unit, see Eq. (2.92) on page 29. Therefore Ty ax is equivalent to the dimensionless potential range Prange of the LSV simulation and Z ax becomes 

The unequal grid was generated with the aid of the Fortran function EE FAC (Appendix E and described in Sect. 7.2). One needs to decide the numbers of points in each of the three ranges, and the minimum intervals, whereupon ee fac produces the required y values for the expansion. One range goes from / = 1 backwards to R = 0. In the other direction, the expansion finds the final variable point at Rmax- The third range is simply 0 Z Z ax- 

---

As mentioned above, Rudolph [478] pointed out that this discretisation yields very poor values and ultimately to poor simulation performance, compared to direct discretisation on an uneven grid, see below. Tests show that particularly at small X values, near the electrode where the greatest changes occur, the second spatial derivatives as seen in (7.7) are approximated very poorly. Rudolph [479,480] and Bieniasz [107] showed that if what we might call the semi-transformed (7.1) is used, rather than the hilly transformed equation, this problem is eliminated. Doing this in a consistent manner, and assuming general transformation functions f(X) and g(Y), we can write for the ith point the approximation... 

As for the choice between direct discretisation on an arbitrarily spaced grid or the formulae for the semi-transformed or the transformed diffusion equation, the present author now inclines towards the first of these. Formulae for the derivatives on arbitrarily spaced points are given in Chap. 3 and Appendix A, and the general subroutine U DERIV is referred to in Appendix C. 

In general, the finding of Rudolph [478], that in one-dimensional simulations, direct discretisation on an unequally spaced grid, rather than equal spacing on a transformed grid, is best, does not appear to apply to UME simulations. Gavaghan made a very thorough study of UMDE simulations... 

In the case of the ultramicroelectrodes such as the disk electrode, it is necessary to integrate over the surface, and sometimes there will be unequally spaced points along the surface, as for example, in direct discretisation on an unequal grid in the example program UME DIRECT. As mentioned in Chap. 12, it is found that due to the errors in the computed concentration values, the local fluxes are so inaccurate that any integration method better than the simple trapezium method is not justified. The routine U TRAP is thus recommended here. It integrates local current densities, precalculated by using the above routine U DERIV. 

As for the choice between direct discretisation on an arbitrarily spaced grid or the formulae for the semi-transformed or the transformed diffusion equation, the present... 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

In this study, the Boltzmann equation is solved with the help of a single relaxation time collision operator approximated by the Bhatnagar-Gross-Krook (BGK) approach [1], Here, the relaxation of the distribution function to an equilibrium distribution is supposed to occur at a constant relaxation parameter r. The substitution of the continuous velocities in the Boltzmann equation by discrete ones leads to the discrete Boltzmann equation, where fai = fm(x, t). The number of available discrete velocity directions ai that connect the lattice nodes with each other depends on the applied model. In this work, the D3Q19 model is used which applies for a three-dimensional grid and provides 19 distinct propagation directions. Discretising time and space with At and Ax = At yields the Lattice-Boltzmann equation ... 

---