Finite-difference electrochemical kinetic simulations using the Rosenbrock time integration scheme

Abstract

Abstract Temporal integration based on multistep backward differentiation formulae may not be the best choice for finite-difference electrochemical kinetic simulations. A one-step, highly accurate, unconditionally stable and non-oscillatory implicit time scheme that also provides temporal error estimates, would be more desirable. The Lawson–Morris–Gourlay extrapolation (LMGE) method, suggested by Strutwolf and Schoeller, partially satisfies these requirements, but it is computationally rather expensive, especially for nonlinear governing equations. The Rosenbrock method, thus far unused in electrochemical simulations, is shown here to be competitive. By combining a three-stage ROWDA3 Rosenbrock method for differential–algebraic equations (DAEs), with a finite-difference spatial discretization of the governing equations, an effective simulation algorithm can be obtained. This is demonstrated for three example kinetic models represented by a single reaction–diffusion equation in one space dimension: potential step experiment, linear sweep voltammetry for a first- and second-order irreversible heterogeneous charge transfer, and linear sweep voltammetry for an enzymatic catalytic reaction mechanism. Numerical properties of the algorithm are compared in calculations with those of the finite-difference methods based on first- and second-order backward differentiation formulae, the Crank–Nicolson (CN) method, and the second-order LMGE method. Fixed, uniform grids are used. Under conditions of a large temporal-to-spatial grid step ratio, important for electrochemical simulations, the third-order accurate ROWDA3 integration proves more efficient than the second-order extrapolation method. In one case it is also more efficient than the method based on the second-order (two-step) backward differentiation formula.