One-dimensional Space Geometry Research Articles

Utility of super-time-stepping for electroanalytical digital simulations by explicit finite difference methods. Part 1: Spatially one-dimensional models

Super-time-stepping denotes a class of special strategies of choosing nonuniform time steps in finite-difference solutions of partial differential equations by conditionally stable explicit techniques, that allow one to effectively overcome limits of numerical stability. This is achieved by demanding the stability only at ends of certain sequences of time steps, called supersteps. The utility of one variant of the super-time-stepping is examined, for digital simulations of electroanalytical experiments described by reaction-diffusion partial differential equations in one-dimensional space geometry. The tests focus on example models of chronoamperometry and cyclic voltammetry. Reductions of computational times, and improvements of the efficiency of such simulations are observed. Implementation of the super-time-stepping requires only minor changes in computer programs for conventional explicit simulation methods. Super-time-stepping can be combined both with sequential and parallel computations, but it should be of particular interest in connection with the efforts towards parallelisation of electroanalytical simulations.

A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) spatial grids in one-dimensional space geometry.

The validity for finite-difference electrochemical kinetic simulations, of the extension of the Numerov discretization designed by Chawla and Katti [J Comput Appl Math 1980, 6, 189-196] for the solution of two-point boundary value problems in ordinary differential equations, is examined. The discretization is adapted to systems of time-dependent reaction-diffusion partial differential equations in one-dimensional space geometry, on nonuniform space grids resulting from coordinate transformations. The equations must not involve first spatial derivatives of the unknowns. Relevant discrete formulae are outlined and tested in calculations on two example kinetic models. The models describe potential step chronoamperometry under limiting current conditions, for the catalytic EC, and Reinert-Berg CE reaction mechanisms. Exponentially expanding space grid is used. The discretization considered proves the most accurate and efficient, out of all the three-point finite-difference discretizations on such grids, that have been used thus far in electrochemical kinetics. Therefore, it can be recommended as a method of choice.

Improving the accuracy of the spatial discretization in finite-difference electrochemical kinetic simulations, by means of the extended Numerov method.

The fourth-order accurate, three-point finite-difference Numerov spatial discretization provides accurate and efficient solutions to the time-dependent governing differential equations of electrochemical kinetics in one-dimensional space geometry, when the equations contain first time derivatives of the solution, second spatial derivatives, and homogeneous reaction terms only. However, the original Numerov discretization is not applicable when the governing equations involve first spatial derivative terms. To overcome this limitation, an appropriately extended Numerov discretization is required. We examine the utility of one of such extensions, first described by Chawla. Relevant discrete formulae are outlined for systems of linear governing equations involving first derivative terms, and applied to five representative example models of electrochemical transient experiments. The extended Numerov discretization proves to have an accuracy and efficiency comparable to the original Numerov scheme, and its accuracy is typically up to four orders of magnitude higher, compared to the conventional, second-order accurate spatial discretization, commonly used in electrochemistry. This results in a considerable improvement of efficiency. Therefore, the application of the extended Numerov discretization to the electrochemical kinetic simulations can be fully recommended.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 15: patch-adaptive simulation of example transient experiments described by Nernst–Planck–electroneutrality equations in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 of the present series of papers, with extensions described in Part 14, is applied to five typical examples of time-dependent models of transient experiments represented by Nernst–Planck–electroneutrality equation systems in one-dimensional space geometry. The models describe: potential step chronoamperometry for a charge neutralisation reaction at a planar electrode in the absence of the supporting electrolyte; potential step chronoamperometry for a planar electrode coated with a polymer film under conditions of migration–diffusion of redox ions; potential step chronoamperometry of uncharged species at a hemispherical electrode in the presence of a low concentration of supporting electrolyte; cyclic voltammetry for an electrode reaction (initially at equilibrium) involving uncharged species, at a planar electrode in the absence of the supporting electrolyte; and current step chronopotentiometry for a liquid|liquid junction between two binary electrolytes. The calculations lead to analogous conclusions, with regard to the performance of the strategy, as in the case of the models represented by the Nernst–Planck–Poisson equations solved in Part 14. New observations reveal that highly non-linear boundary conditions present an additional challenge for the Newton iterative method utilised by the strategy. The ability of the strategy to solve non-local boundary conditions expressing current equality at distant boundaries is also demonstrated. However, formulating such boundary conditions correctly is not straightforward, so that their use is not recommended. Adaptive solutions suggest inaccuracies in the literature solutions for two example models.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 14: extension of the patch-adaptive strategy to time-dependent models involving migration–diffusion transport in one-dimensional space geometry, and its application to example transient experiments described by Nernst–Planck–Poisson equations

The finite-difference patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 and extended in Part 10 of this series of papers, is further extended to time-dependent models involving migration–diffusion transport in one-dimensional space geometry. The extensions include: spatial discretisation of generalised second spatial derivative expressions typical of migration–diffusion equations; allowance for the dependence of boundary conditions on displacement current; support for an a posteriori calculation of the displacement current as one of the model responses; optional calculation of steady-state initial conditions; the ability to use enhanced precision of floating point calculations. The extended strategy is used to simulate four examples of transient experiments, represented by Nernst–Planck–Poisson equation systems: coulostatic charge injection at an ideally polarised planar electrode; a voltage step for a thin layer asymmetric electrochemical cell; chronopotentiometry for an electrolyte|membrane|electrolyte system; and chronopotentiometry for a bipolar membrane. The strategy provides fairly reliable solutions, but its automatism and efficiency are less satisfactory compared to models without electric migration, owing to the need for model-dependent tuning of the method parameters, and increased computational cost necessary for the exact adaptive determination of the electric potential profiles.

High order accurate, one-sided finite-difference approximations to concentration gradients at the boundaries, for the simulation of electrochemical reaction-diffusion problems in one-dimensional space geometry

Accurate calculation of concentration gradients at the boundaries is crucial in electrochemical kinetic simulations, owing to the frequent occurrence of gradient-dependent boundary conditions, and the importance of the gradient-dependent electric current. By using the information about higher spatial derivatives of the concentrations, contained in the time-dependent, kinetic reaction-diffusion partial differential equation(s) in one-dimensional space geometry, under appropriate assumptions it is possible to increase the accuracy orders of the conventional, one-sided n-point finite-difference formulae for the concentration gradients at the boundaries, without increasing n. In this way a new class of high order accurate gradient approximations is derived, and tested in simulations of potential-step chronoamperometric and current-step chronopotentiometric transients for the Reinert–Berg system. The new formulae possess advantages over the conventional gradient approximations. For example, they allow one to obtain a third order accuracy by using two space points only, or fourth order accuracy by using three points, and yet they yield smaller errors than the conventional four-point, or five-point formulae, respectively. Needing fewer points, for approximating the gradients with a given accuracy, simplifies also the solution of the linear algebraic equations arising from the application of implicit time integration schemes.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 12. Patch-adaptive simulation of example transient experiments described by kinetic models defined over multiple space intervals in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, with extensions described in Part 10, is applied to three representative examples of time-dependent kinetic models defined over multiple space intervals in one-dimensional space geometry. The models describe: cyclic voltammetry for a diffusion-controlled electron transfer reaction at a liquid ∣ liquid interface; linear potential sweep voltammetry for a diffusion-controlled charge transfer at a planar amalgam film electrode; and potential step chronoamperometry for a free monoenzymatic amperometric biosensor. The strategy provides accurate, efficient and nearly automatic solutions for these examples, which confirms the overall adequacy of the extensions of the strategy, associated with the handling of multiple space intervals. However, limiting cases of the models, characterised by disparate lengths of the spatial intervals, appear difficult to solve. A possible occurrence of an error in the literature simulation results for the biosensor example is also revealed.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 11. Patch-adaptive simulation of example transient experiments described by kinetic models involving simultaneously distributed and localised unknowns, in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, with extensions described in Part 10, is applied to three representative examples of time-dependent kinetic models in one-dimensional space geometry, characterised by the simultaneous presence of spatially distributed and localised unknowns. The models describe: potential step chronoamperometry for a diffusion-controlled charge transfer reaction accompanied by the adsorption of reactants at the electrode; cyclic voltammetry for an electrochemical deposition–precipitation mechanism; and double potential step chronoamperometry for an oscillatory system involving a diffusion-controlled charge transfer reaction having a negative polarisation resistance, in the presence of double layer charging, and ohmic potential drops. The strategy provides accurate, efficient and nearly automatic solutions for these examples, which confirms the overall adequacy of the extensions of the strategy, associated with the handling of localised unknowns. In the third example, the adaptive simulations are very detailed and informative. They confirm the occurrence of numerical artefacts in the literature results of fixed grid simulations for this example.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 10. Extension of the patch-adaptive strategy to kinetic models involving spatially localised unknowns at the boundaries, multiple space intervals, and non-local boundary conditions, in one-dimensional space geometry

The finite-difference patch-adaptive strategy for electrochemical kinetic simulations, described in Part 5 of this series of papers, is extended to time-dependent models in one-dimensional space geometry, involving spatially localised unknowns at the boundaries, multiple space intervals, and non-local boundary conditions. These extensions are of interest, among others, for the modelling studies of: electrochemical adsorption and/or electrocatalytic reactions, liquid ∣ liquid interfaces, amalgam electrodes, electrochemical biosensors, and pattern formation at electrodes. The following new features are included in the strategy. Grid objects, used by the spatial grid adaptation algorithm, are decomposed into interior and boundary subgrid objects, keeping discrete values of the spatially distributed and localised unknowns, respectively. Linear algebraic equation systems, characterised by quasi-block-tridiagonal matrices having variable block dimensions, and isolated block-pentadiagonal rows, are solved by a specially developed variant of the Thomas algorithm, previously published by the present author. Relative error control is applied to localised unknowns. The calculation of consistent boundary solutions takes into account temporal derivatives present in boundary conditions. Accurate a posteriori calculation of temporal derivatives of the localised unknowns is enabled.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Patch-adaptive simulation of moving fronts in non-linear diffusion models of the switching of conductive polymers

The finite-difference patch-adaptive strategy (PAS) for electrochemical kinetic simulations, previously described by the author, is extended and applied to two examples of non-linear diffusion in one-dimensional space geometry, characterised by moving fronts, and related to the modelling of redox switching of conductive polymers. The extension consists of an appropriate spatial discretisation of the second derivative diffusion term involving concentration-dependent diffusion coefficient, and of an improved selection of starting approximations for the Newton iterations within the extrapolation time-stepping scheme. Adaptive solutions obtained are reliable, accurate, efficient, and superior to fixed-grid solutions.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

The patch-adaptive strategy described in Part 5 proves to provide correct, fairly efficient, and nearly automatic solutions to six representative example electrochemical kinetic models in one-dimensional space geometry, that exhibit difficult-to-resolve local spatial and temporal solution structures at the electrodes. The models describe: the square-wave controlled potential transient for an uncomplicated heterogeneous charge transfer reaction, the potential step method and linear potential sweep voltammetry for the standard catalytic reaction mechanism with a pseudo-first-order homogeneous reaction, linear potential sweep voltammetry for an EC reaction mechanism involving a follow-up homogeneous dimerisation, steady-state voltammetry at a rotating disc electrode, and steady-state voltammetry at a spherical microelectrode. These examples are characterised by extremely thin reaction layers and other instances of local spatial regions of considerable variations of the solution, including transient effects caused by temporal discontinuities of boundary conditions. The strategy creates dynamically spatio-temporal grids well adapted to such solution structures, without any a priori knowledge about their location. The accuracy of the numerical solutions proves well controlled by means of error tolerance parameters. Numerical difficulties observed with the previously described adaptive moving grid technique do not occur in the present strategy. However, further work is needed to improve the performance of the strategy in the cases of thin reaction layers associated with fast second-order homogeneous reactions, and boundary layers associated with hydrodynamic electroanalytical methods, for which the computational effort appears too large.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

Abstract 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.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

The patch-adaptive strategy described in Part 5 proves to provide correct, fairly efficient, and nearly automatic solutions to three representative example electrochemical kinetic models in one-dimensional space geometry, that exhibit difficult-to-resolve moving reaction fronts in the electrolyte, away from the electrode. The models describe: double potential step experiments for a simple mechanism of electrochemically generated luminescence, linear potential sweep voltammetry for an EE-DISP mechanism with Nernstian charge transfers and a reversible homogeneous disproportionation reaction, and linear potential sweep voltammetry for an RRC-initiated electropolymerization reaction scheme. The strategy dynamically creates spatio-temporal grids that adaptively concentrate in the regions of the reaction fronts, without any a priori knowledge about their location. Numerical difficulties, such as electric current oscillations observed with the previously described adaptive moving grid technique in the case of moving reaction fronts, do not occur in the present strategy. However, further work is needed to improve the performance of the strategy in the cases of reaction fronts associated with fast and irreversible second-order homogeneous reactions, for which the computational costs are too large. Improvements are also desirable in the case of the simulation of a rapidly moving reaction front occurring in the electropolymerization model, for which the strategy tends to provide incorrect front positions.

Excitable wave patterns in a spatially extended nonlinear optical cavity

We show that finite external excitation can lead to a traveling wave in an excitable passive optical system with one-dimensional space geometry. We have studied the excitable behavior of this system in parallel with that of its diffusive counterpart and show the effects of optical phase on the traveling-wave solution and its velocity. In two-dimensional space we observe numerically rotating optical spiral waves evolving from a truncated planar wave front.

ELSIM—a problem-solving environment for electrochemical kinetic simulations. Version 3.0-solution of governing equations associated with interfacial species, independent of spatial coordinates or in one-dimensional space geometry

A third version of the program ELSIM (for IBM compatible PCs) for the simulation of electrochemical transient methods has been elaborated upon. The program has been equipped with numerical algorithms that allow solution of kinetic problems characterized by the presence of interfacial species. Problems of this kind play an important role in electrochemical kinetic studies, among others in connection with electrocatalysis, modified electrodes or oscillatory systems. The user-written governing equations may take the form of differential—algebraic equations (for the concentrations of interfacial species) independent of spatial coordinates, or partial differential equations (for the concentrations of bulk species) in one-dimensional space geometry, coupled with differential—algebraic equations. Problem formulation has also been facilitated by the inclusion of a reaction compiler, which automatically generates the text of the above governing equations for user-written reaction mechanisms. Other new features include an option for the least-squares fitting of simulated transients to experimental curves, a hypertext help facility, and enhanced performance owing to the revised formula translation based on the three-address internal code generation, and on automatic differentiation.

A reaction compiler for electrochemical kinetics

A computer code serving for an automatic translation of user-written source texts of electrochemical reaction mechanisms into corresponding target texts of mathematical equations that govern the kinetics of electrochemical systems under transient conditions is reported. The rules of the language enabling symbolic specification of the reaction mechanisms, the compiler options, and conventions regarding the target formulae are outlined and illustrated by examples. A considerable diversity of reaction mechanisms involving equilibrium, non-equilibrium reversible or irreversible reactions that can be electrochemical, heterogeneous non-electrochemical or homogeneous, is permitted. The reactions may involve bulk species (distributed in the electrolyte volume) and interfacial species (localized at the electrodes) of variable or constant concentrations, and electrons. The transient conditions may correspond to a number of electrochemical techniques, including potential-step method, linear potential scan voltammetry and chronopotentiometry. For kinetic problems in one-dimensional space geometry the generated governing equations take the general form of the reaction-advection-diffusion partial differential equations for the concentrations of bulk species (with initial and boundary conditions), optionally coupled with algebraic, ordinary differential or differential-algebraic equations for the concentrations of interfacial species. The governing equations can be obtained in the form of ELSIM problem definitions, enabling further solution by means of this simulation program.