Unequally Spaced Grids

Considering the above aspects, it is beneficial to establish a grid that mimics the concentration profile and provides a high number of points near the electrode surface but minimises the total number, which makes the simulation process more efficient. With this aim, unequally spaced grids or transformation of the spatial coordinates can be employed, the former being preferred in the simulation of electrochemical experiments [3]. 

Exponentially expanding grids are widely employed in the simulation of electrochemical experiments such that the distance between consecutive points of the grid expands exponentially according to the following definition  

Programmatically, it is necessary to calculate the position of every grid point Xi at the start of a simulation. The following code fragment can be used for such a calculation  

The position of every point in the spatial grid is stored in a vector, and the number of spacesteps, n, is now equal to the number of elements in this vector (rather than being calculated ahead of time). 

After introducing unequal spatial intervals, the finite differences must be reformulated. Following the central three-point approximation introduced in Chapter 3, the second derivative can be approximated as 

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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... 

In 1924, the Russian astronomer Numerov (transliterating his own name as Noumerov), published a paper [421] in which he described some improvements in approximations to derivatives, to help with numerical simulations of the movement of bodies in the solar system. His device has been adapted to the solution of pdes, and was introduced to electrochemistry by Bieniasz in 2003 [108]. The method described by Bieniasz is also called the Douglas equation in some texts such as that of Smith [514], where a rather clear description of the method is found. With the help of the Numerov method, it is possible to attain fourth order accuracy in the spatial second derivative, while using only the usual three points. The first paper by Bieniasz on this method treated equally spaced grids, and was followed by another on unequally spaced grids [107], The method makes it practical to use higher-order time derivative approximations without the complications of, say, the (6,5)-point scheme described above, which makes the solution of the system of equations a little complicated (and computer time consuming). 

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... 

M. Rudolph. Digital simulations on unequally spaced grids. Part 1. Critical remarks on using the point method by discretisation on a transformed grid, J. Electroanal. Chem. 529, 97-108 (2002). 

Rudolph M (2004) Digital simulations on unequally spaced grids. Part 3. Attaining exponential convergence for the discretisation error of the flux as a new strategy in digital simulations of electrochemical experiments. 1 Electroanal Chem 571 289-307... 

Fig. 12.12 The unequally spaced grid for the three-dimensional simulation of a rectangular electrode with L = 2. (a) XY plane (b) XZ plane...

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. 

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