Discretising the Transformed Equation

Transformation (7.3) leads to the new diffusion (7.5) in Y-space. Although it is fairly obvious how the new right-hand side is discretised, for completeness, this will be described here. 

Instead of a number of sample points in X, we now have a number of equally spaced points along the new coordinate Y with a spacing of 6Y. Without considering which simulation algorithm is to be used, we discretise the new (7.5) at the point Y.j as follows  

The coefficients in the right-hand term in brackets in (7.8) can be precomputed, as can the row of values. Further details of how all this is implemented are given in Chap. 8 for the respective simulation algorithms. 

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 

Using the transformation (7.3) and thus substituting for g (Y) as given in (7.4) at the indices given and rearranging a little, this becomes 

---