There are times when an electrochemical experiment requires reasonably accurate knowledge of doub... more There are times when an electrochemical experiment requires reasonably accurate knowledge of double-layer capacitance c d l and/or uncompensated series resistance R,. An example is the application of iR compensation-one needs to know how much there is to compensate (if R, is constant and an appropriate technique for this is employed). The simplest expedient is t o increase the level of compensation until the potentiostat/cell system is no longer stable. In a recent review ( I ) , it was pointed out that this can be a rather inaccurate procedure, as many systems may be unstable a t rather less than 100% compensation and, conversely, some are stable even beyond 100%. T h e alternative of carrying out highly accurate measurements using a bridge ( 2 ) or by impedance phase-angle and -magnitude, either by hand ( 3 ) or by means of suitable (and often expensive and otherwise perhaps not needed) instruments may be too costly and time-consuming and unnecessarily accurate.
An important historical paper on the numerical solution of pde's has regularly, but incorrectly, ... more An important historical paper on the numerical solution of pde's has regularly, but incorrectly, been assigned to the year 1951. The origin of this error of reference is discussed.
ABSTRACT The mathematical analysis of the consistency of Feldberg's simple BDF start in e... more ABSTRACT The mathematical analysis of the consistency of Feldberg's simple BDF start in electrochemical digital simulation is presented. The method leads to rather accurate results compared with the more obvious rational start in which the BDF order is worked up from a two-point start, increasing the number of points in time until the desired number is reached. It is proved that the simple start used by the Feldberg school, combined with the subtraction of half a time interval from all nominal times during the simulation is mathematically consistent.
.ECTURE NOT IN PHYSICS D. Britz ES Digital Simulation in Electrochemistry Third Edition 4^ Sprin ... more .ECTURE NOT IN PHYSICS D. Britz ES Digital Simulation in Electrochemistry Third Edition 4^ Sprin ... The Editorial Policy for Monographs The series Lecture Notes in Physics reports new developments in physical research and teaching - quickly, informally, and at a high level. ...
Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 1968
... inorganic species (with the exception of thallium, for which anomalous behaviour was noted by... more ... inorganic species (with the exception of thallium, for which anomalous behaviour was noted by TAMAMUSHI AND TANAKA1 and SALIE AND LORENZO). ... small in magnitude compared with that of the double-layer capacity (coC~) at sufficiently high frequencies, ie, lim (Yr/coCR ...
The application of fourth-order finite difference discretisations of the second derivative of con... more The application of fourth-order finite difference discretisations of the second derivative of concentration with respect to distance from the electrode, in electrochemical digital simulations, is examined further. In the bulk of the diffusion space, a central 5-point scheme is used, and 6-point asymmetric schemes are used at the edges. In this paper, four Runge-Kutta schemes have been used for the time integration. The observed efficiencies, for the Cottrell experiment and chronopotentiometry, are satisfactory, going beyond those for the 3-point scheme. However, it is third-order Runge-Kutta, rather than the fourth-order scheme, which is the most efficient, the two resulting in practically the same errors. This is probably due to the computational procedure where a constant ratio of dt/h 2 was used. : S 0 0 9 7 -8 4 8 5 ( 0 1 ) 0 0 0 8 6 -9
The Crank Á/Nicolson (CN) simulation method has an oscillatory response to sharp initial transien... more The Crank Á/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. #
ABSTRACT We extend the analysis of the stepwise numerical stability of the classic explicit, full... more ABSTRACT We extend the analysis of the stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference algorithms for electrochemical kinetic simulations, to the multipoint gradient approximations at the electrode. The discussion is based on the matrix method of stability analysis.
ABSTRACT The stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicol... more 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.
The stepwise numerical stability of the Saul'yev finite difference discretization of ... more The stepwise numerical stability of the Saul'yev finite difference discretization of an example diffusional initial boundary value problem 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 on stability, assuming the two-point, forward-difference approximation for the gradient at the
There are times when an electrochemical experiment requires reasonably accurate knowledge of doub... more There are times when an electrochemical experiment requires reasonably accurate knowledge of double-layer capacitance c d l and/or uncompensated series resistance R,. An example is the application of iR compensation-one needs to know how much there is to compensate (if R, is constant and an appropriate technique for this is employed). The simplest expedient is t o increase the level of compensation until the potentiostat/cell system is no longer stable. In a recent review ( I ) , it was pointed out that this can be a rather inaccurate procedure, as many systems may be unstable a t rather less than 100% compensation and, conversely, some are stable even beyond 100%. T h e alternative of carrying out highly accurate measurements using a bridge ( 2 ) or by impedance phase-angle and -magnitude, either by hand ( 3 ) or by means of suitable (and often expensive and otherwise perhaps not needed) instruments may be too costly and time-consuming and unnecessarily accurate.
An important historical paper on the numerical solution of pde's has regularly, but incorrectly, ... more An important historical paper on the numerical solution of pde's has regularly, but incorrectly, been assigned to the year 1951. The origin of this error of reference is discussed.
ABSTRACT The mathematical analysis of the consistency of Feldberg's simple BDF start in e... more ABSTRACT The mathematical analysis of the consistency of Feldberg's simple BDF start in electrochemical digital simulation is presented. The method leads to rather accurate results compared with the more obvious rational start in which the BDF order is worked up from a two-point start, increasing the number of points in time until the desired number is reached. It is proved that the simple start used by the Feldberg school, combined with the subtraction of half a time interval from all nominal times during the simulation is mathematically consistent.
.ECTURE NOT IN PHYSICS D. Britz ES Digital Simulation in Electrochemistry Third Edition 4^ Sprin ... more .ECTURE NOT IN PHYSICS D. Britz ES Digital Simulation in Electrochemistry Third Edition 4^ Sprin ... The Editorial Policy for Monographs The series Lecture Notes in Physics reports new developments in physical research and teaching - quickly, informally, and at a high level. ...
Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 1968
... inorganic species (with the exception of thallium, for which anomalous behaviour was noted by... more ... inorganic species (with the exception of thallium, for which anomalous behaviour was noted by TAMAMUSHI AND TANAKA1 and SALIE AND LORENZO). ... small in magnitude compared with that of the double-layer capacity (coC~) at sufficiently high frequencies, ie, lim (Yr/coCR ...
The application of fourth-order finite difference discretisations of the second derivative of con... more The application of fourth-order finite difference discretisations of the second derivative of concentration with respect to distance from the electrode, in electrochemical digital simulations, is examined further. In the bulk of the diffusion space, a central 5-point scheme is used, and 6-point asymmetric schemes are used at the edges. In this paper, four Runge-Kutta schemes have been used for the time integration. The observed efficiencies, for the Cottrell experiment and chronopotentiometry, are satisfactory, going beyond those for the 3-point scheme. However, it is third-order Runge-Kutta, rather than the fourth-order scheme, which is the most efficient, the two resulting in practically the same errors. This is probably due to the computational procedure where a constant ratio of dt/h 2 was used. : S 0 0 9 7 -8 4 8 5 ( 0 1 ) 0 0 0 8 6 -9
The Crank Á/Nicolson (CN) simulation method has an oscillatory response to sharp initial transien... more The Crank Á/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. #
ABSTRACT We extend the analysis of the stepwise numerical stability of the classic explicit, full... more ABSTRACT We extend the analysis of the stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference algorithms for electrochemical kinetic simulations, to the multipoint gradient approximations at the electrode. The discussion is based on the matrix method of stability analysis.
ABSTRACT The stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicol... more 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.
The stepwise numerical stability of the Saul'yev finite difference discretization of ... more The stepwise numerical stability of the Saul'yev finite difference discretization of an example diffusional initial boundary value problem 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 on stability, assuming the two-point, forward-difference approximation for the gradient at the
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