Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations.: Part 13. Patch-adaptive simulation of wave propagation along ring electrodes: one-dimensional approximation

The patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 of this series of papers, is used to simulate the propagation of potential and concentration waves along ring electrodes, under galvanostatic control. The simulation is based on the one-dimensional model of Koper and Sluyters, combined with periodic boundary conditions adequate for ring electrodes. In this way, the extension of the strategy to non-local boundary conditions, described in Part 10 of this series of papers, is tested in the presence of a difficult-to-solve moving front crossing the boundaries with the periodic boundary conditions. The validity of the strategy for simulating this kind of pattern formation at electrodes is confirmed by comparisons with analytical solutions for the asymptotic case of the model. The adaptive simulations reveal that close to the asymptotic conditions, the initial, fast relaxation of the propagating waves of the electric potential drop across the double layer, does not immediately result in a steady-state, but it is followed by a slow decay of the remaining, non-moving spatial inhomogeneities of the concentration and potential profiles along the ring electrode.