Digital simulations on unequally spaced grids.: Part 1. Critical remarks on using the point method by discretisation on a transformed grid

Abstract

The paper demonstrates that the variable transformation variant of the point method (point 2 methods throughout the paper) suffers from serious problems, which prevent efficient simulations of electrochemical kinetic–diffusion systems. It will be shown that the limitation of the applicability of this method to very slowly expanding grids and small values of the grid parameter a does not result from the nature of the diffusion problem. It is simply a consequence of inaccuracies introduced by discretizing equations resulting from a variable transformation which hold true only if the limiting process Δ Y→0 is really executed. Such problems can be avoided by an alternative implementation of the point method (point 1 method throughout the paper) based on discretizing the second-order space derivative directly on the expanding grid using unequal intervals. This method is only first-order accurate but works much better than the variable transformation variant. It gives fairly accurate results on moderately expanding grids (Δ Y≤0.25) no matter how large the parameter a is. However, due to the reduced precision of both methods neither of the two is able to compete with more accurate techniques such as the finite element or box method.