Dieter Britz

Investigation of the relative merits of some n-point current approximations in digital simulation : Application to an improved implicit algorthm for Quasi-reversible systems

Abstract The n-point current approximation (n = 2, 3, …) in electrochemical digital simulation are examined for accuracy by using two model concentration profiles, both for point and box spacing. The recommendation is n = 5. This finding is incorporated in a simplified and generalized implicit flux scheme for simulating quasi-reversible systems.

Damping of Crank-Nicolson error oscillations

The Crank-Nicolson (CN) simulation method has an oscillatory response to sharp initial transients. The technique is convenient but the oscillations make it less popular. Several ways of damping the oscillations in two types of electrochemical computations are investigated. For a simple one-dimensional system with an initial singularity, subdivision of the first time interval into a number of equal subintervals (the Pearson method) works rather well, and so does division with exponentially increasing subintervals, where however an optimum expansion parameter must be found. This method can be computationally more expensive with some systems. The simple device of starting with one backward implicit (BI, or Laasonen) step does damp the oscillations, but not always sufficiently. For electrochemical microdisk simulations which are two-dimensional in space and using CN, the use of a first BI step is much more effective and is recommended. Division into subintervals is also effective, and again, both the Pearson method and exponentially increasing subintervals methods are effective here. Exponentially increasing subintervals are often considerably more expensive computationally. Expanding intervals over the whole simulation period, although capable of satisfactory results, for most systems will require more cpu time compared with subdivision of the first interval only.

Implicit calculation of boundary values in digital simulation applied to several types of electrochemical experiment

Abstract In the digital simulation of chronopotentiometry, potentiodynamic techniques and adsorption kinetics involving both diffusion and adsorption isotherms, the problem of species boundary concentrations at a given new time step can be eliminated by including these in the implicit equation systems for the Crank-Nicolson scheme. This greatly improves simulation efficiency. The respective expressions for the various types of experiment are developed and tested.

Kinetics of the deuterium and hydrogen evolution reactions at palladium in alkaline solution

Abstract The kinetics of the deuterium evolution reaction (DER) at palladium in alkaline solution were investigated by measuring the overpotential decay transients arising from interruption of the polarising current. It was found that the overpotential η could be time-resolved into two separate components. The initial component η 1 arises from the Volmer reaction, while the longer-lived component η 2 is attributed to the Tafel reaction. The Volmer-Tafel mechanism was also confirmed by examining the dependence of η 2 on η, although there were some indications that, at high overpotentials, the Heyrovsky reaction is also involved. Studies of the hydrogen evolution reaction (HER) at palladium were also consistent with a Volmer-Tafel process. The rates of the Volmer and Tafel reactions were found to be higher for the HER than the DER, in agreement with known electrolytic H/D separation factors. Measurements of η 2 were used to estimate the stoichiometry of the deuterated palladium cathode using the concept of equivalent D 2 pressure. The D/Pd loading ratios were found to increase with current density, reaching values of between 0.78 and 0.82 at 50 m A cm −2 .

Electrochemical digital simulation by runge-kutta integration

Abstract An explicit technique for the digital simulation of electrochemical dynamics, more efficient than classical finite differences and easily implemented and modified, is described. It is Runge-Kutta integration of the system of first-order differential equations, obtained by semi-discretizing the second-order partial differential equations for a given system. The method allows somewhat larger rate constant values in the simulation of systems involving homogeneous chemical reactions, compared with the standard explicit method, although it cannot handle extremely large rate constants, which result in very small concentrations or a compressed reaction layer. It is found that, in general, the second-order Runge-Kutta formula is sufficient although, for moderately large chemical rate constants, the third-order formula may be appropriate.

Electrochemical digital simulation: incorporation of the Crank—Nicolson scheme and n-point boundary expression into the Rudolph algorithm

Abstract The Rudolph algorithm for solving coupled multispecies mechanisms was described with the Laasonen (backward-implicit) scheme and two-point boundary expressions. This paper uses the Rudolph approach but with the more accurate Crank—Nicolson scheme and general n -point boundary expressions. Example computations are reported for a simple catalytic mechanism (potential jump) and a more complex second-order catalytic mechanism (LSV).

Some numerical investigations of the stability of electrochemical digital simulation, particularly as affected by first-order homogeneous reactions

Abstract A reported analysis of the stability of some digital simulation methods is investigated by numerical experiments and the results are consistent with the analysis. Traditional stability conditions need to be modified slightly in the presence of homogeneous reactions, though not to a degree that has practical significance. The oscillatory response of the Crank-Nicolson method compared with the Laasonen method can be reduced or eliminated by a preliminary expansion of the time steps, as suggested by Feldberg.

Higher-order spatial discretisations in electrochemical digital simulations. Part 4. Discretisation on an arbitrarily spaced grid

A systematic approach to the construction of finite difference formulae for the approximation to first and second derivatives with respect to space on an arbitrarily spaced grid is presented. The finite difference formulae, combined with backward implicit (BI) and extrapolation methods, are used for electrochemical simulations and tested for efficiency. Excellent results are obtained with second and third order discretisations even for very small space intervals in the vicinity of the electrode and strongly expanding grid spacings. This ensures efficient simulation of kinetic-diffusion systems where, due to a fast homogeneous reaction, a thin reaction layer is formed adjacent to the electrode.

High-order spatial discretisations in electrochemical digital simulation. 1. Combination with the BDF algorithm

The application of fourth-order discretisations of the second derivative of concentration with respect to distance from the electrode, in electrochemical digital simulations, is examined. In the bulk of the diffusion space, a central five-point scheme is used, and six-point asymmetric schemes are used at the edges. In this paper, the scheme is applied to the BDF technique which allows higher orders in time as well. The method is found to be stable, using both the Neumann and matrix methods. Performance with BDF is not, however, optimal, levelling off at three-point BDF, as does the usual three-point approximation. This is shown to be due to startup problems inherent with BDF.

Numerical stability of finite difference algorithms for electrochemical kinetic simulations: Matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods and typical problems involving mixed boundary conditions

Abstract The stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference discretizations of example diffusional initial boundary value problems from electrochemical kinetics has been investigated using the matrix method of stability analysis. Special attention has been paid to the effect of the discretization of the mixed, linear boundary condition with time-dependent coefficients on stability, assuming the two-point forward-difference approximations for the gradient at the left boundary (electrode). Under accepted assumptions one obtains the usual stability criteria for the classic explicit and fully implicit methods. The Crank-Nicolson method turns out to be only conditionally stable in contrast to the current thought regarding this method.