Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 5. A finite-difference, adaptive space/time grid strategy based on a patch-type local uniform spatial grid refinement, for kinetic models in one-dimens

Local, difficult-to-solve spatio-temporal structures such as extremely thin reaction layers at the electrodes, thin moving reaction fronts, fast transient variations, isolated temporal discontinuities in boundary conditions, edge effects, etc. are characteristic of the initial boundary value and boundary value problems occurring in electrochemical kinetic modelling. Further progress in the simulation methodology is hindered by the lack of automatic solution techniques for such problems. Based on a critical consideration of various adaptive grid strategies for partial differential equations reported in the scientific literature, a new finite-difference adaptive grid strategy has been formulated, that is especially designed for electrochemical kinetic simulations, and is currently limited to models in one-dimensional space geometry. The intention is to enable an automatic solution of the governing partial/ordinary differential equations to a prescribed accuracy, without any a priori knowledge about the spatio-temporal location of the emerging solution structures. In view of the importance of the solution (concentration) gradients for the electrochemical theory, simultaneous control of the spatial errors of the solutions and their spatial gradients is included in the strategy. Spatial grid adaptation is based on a local uniform grid refinement, using overlapping grid patches. Temporal step selection uses a recent control theoretic algorithm, combined with a simple method of detecting temporal discontinuities of the boundary conditions. A third-order accurate, implicit and L-stable Rosenbrock time-stepping scheme ROWDA3 is used to enable efficient and non-oscillatory temporal integration in the time intervals where a regular solution occurs. A second-order accurate Lawson–Morris–Gourlay extrapolation scheme is used to enable a reliable time-stepping at temporal discontinuities.