Electrochemical digital simulations with an exponentially expanding grid: General expressions for higher order approximations to spatial derivatives: The special case of four-point formulas and their application to multipulse techniques in planar and any size spherical electrodes

Abstract

This paper deals with the consideration, in digital simulation, of a grid in which there is a geometrical relationship between the distances between neighbouring pairs of points, leading to an exponentially expanding sequence of intervals. We show that the expressions for derivatives for higher order spatial discretisations take a very simple form, with coefficients which are dependent on the number of points taken to the left and right of the point considered, but independent of the absolute position of the point in the grid. This enables convenient and efficient manipulation and incorporation in computer programs of these derivatives in the context of finite difference approach. Moreover, the explicit expressions obtained lead to very interesting particular situations. The most interesting is the optimal behaviour of the four-point spatial discretisation for very high expansion factors, which, in combination with fully implicit time-integration schemes, leads to very fast and accurate calculations both for planar and any size spherical electrodes, including microelectrodes. Applications of these coefficients in the simulation of some multipulse and square wave experiments are also presented.