Orthogonal Collocation OC

The literature is not especially lucid on the subject of OC, including the original Whiting and Carr (1977) paper which is, however, clearer than most. In this section, I shall give what I hope is a clear description but will not go into the fine detail because I feel that OC is less accessible, requiring, for example, a familiarity with matrix algebra and the use of rather sophisticated integration techniques that most electrochemists probably do not have. For those who have, the 

Let us, for simplicity, assume a single substance. Its dimensionless concentration C = time- and space dependent but, as usual, we do not indicate this, to simplify the notation. Assuming some sort of homogeneous chemical reaction, we can write the diffusion equation as 

making X = about 0.1. Thus we have relatively close spacing near the electrode and incidentally also at the outer boundary but we don t care, although Yen and Chapman (1982) have suggested using a set of collocation points spread in a semi-infinite fashion, crowding closely only at the electrode. 

Following Whiting and Carr (1977) we write Eqs. 5.110, 5.115 and 5.116 in the more compact matrix forms, respectively, for all,  

Assuming this done (that is, [V] and [W] are now known), let us continue. 

The finite volume method FVM should probably be studied more than it is for electrochemical applications, but it has been applied [ 161,162]. It is possibly related to the box method. A good text on FVM is that of Patankar [11], 

The newer method of Bortels et al., called multidimensional upwinding method (MDUM) should also be mentioned [163], It was applied to a problem involving diffusion, convection and migration, both steady state and time-marching. 

This is one of the variants of the FEMs. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi or Chebyshev polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their uneven spacing, crowding closer both at the electrode and (perhaps strangely) at the outer limit, and the fact that the outer limit is always unity. This is discussed below. 

The possibly peculiar spacing of the collocation points, crowding close both at the electrode and at the outer diffusion limit, does not matter too much, and seems unnecessary. For example, using only five internal points (that is, five apart from zero and unity), they are placed at the values 0.047, 0.231, 0.5, 0.769, 0.953, a series that is symmetrical about the midway point at 0.500. This spacing has been circumvented by Yen and Chapman [174], using Chebyshev polynomials that open out towards the outer limit. Their work has apparently not been followed up. 

The essence is that, if the concentration profile simulated is smooth (which it normally is), then the polynomials will be well behaved in between points and no such problems will be encountered. As is seen below, implicit boundary values can easily be accommodated, and by the use of spline collocation [179-181], homogeneous chemical reactions of very high rates can be simulated. This refers to the static placement of the points. Having, for example, the above sequence of points for five internal points, the point closest to the electrode is at 0.047. This will be seen, below, to be in fact further from the electrode than it seems, because of the way that distance X is normalised so that, for very fast reactions that lead to a thin reaction layer, there might not be any points within that layer. Spline collocation thus takes the reaction layer and places another polynomial within it, while the region further out has its own polynomial. The two polynomials are designed such that they join smoothly, both with the same gradient at the join. This will not be described further here. 

Here are a few brief references to recent or key works in which these methods have been described as used in electrochemical simulations. The interested reader is urged to look these up and follow the references contained in them to the seminal works and text books. Of necessity, much work is left uncited here. 

Ferrigno et al. [239] describe the use of FEM for steady state simulations of recessed, flush and protruding ultramicrodisk electrodes, giving a good description of FEM. The method was made adaptive by Nann (and Heinze) [407,408], and Harriman et al. later published an extensive series of papers on adaptive FEM [287,288,289,290,291]. 

BEM might be thought of as best suited to steady state problems, and has been used for this, for example in corrosion simulations [64] and current distributions [198], but recently also for time-marching problems [457]. 

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Mathematical methods exist that guarantee an optimal placement of the collocation points. In orthogonal collocation (OC), the collocation points are equal to the zero points of the orthogonal polynomials. 

PARSIM optionally provides the method of Finite Differences (FD) for space discretization. An advantage of this method is the lower bandwidth of the Jacobian matrix. Nevertheless, much more node points are needed to achieve the same accuracy compared to the OCFE method as demonstrated below. The method of global Orthogonal Collocation (OC) is provided additionally by PARSIM but should be used only for systems without steep gradients. 

This work investigates the use of reduced order models of reactive absorption processes. Orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE) are compared. All three methods are able to accurately describe the steady state behaviour, but they predict different dynamics. In particular, the OC dynamic models show large unrealistic oscillations. Balanced truncation, residualization and optimal Hankel singular value approximation are applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy. 

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