Electrochimica Acta 55 (2010) 721728

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Electrochimica Acta

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Automatic simulation of cyclic voltammograms bfor weakly singular second kind Volterra integral

Lesaw Ka Institute of Ph d Cheul. Niezapominb Cracow Unive

a r t i c l

Article history:Received 2 JulReceived in reAccepted 6 SepAvailable onlin

Keywords:ComputationaLaboratory autCyclic voltammDigital simulationSecond kind Volterra integral equationsWeakly singular kernels

toryelectrulatir wose haquatiwea

gralson

theory of cyclic voltammetry. The performance of the method is found satisfactory, although errors ofthe simulated transients may deviate from the prescribed error tolerance parameter, so that achievinga given target accuracy is less straightforward than it was for the voltammograms described by the rstkind Abel equations.

2009 Elsevier Ltd. All rights reserved.

1. Introdu

Modernsurementswithin the rmuch as pocontrol. Moinstrumentthe time ismodelling [analysis asthe creationautonomicbox devicProblem Soporting themodel-buil

Corresponof Sciences, Detion, ul. Niezapfax: +48 12 42

E-mail addURL: http:

0013-4686/$ doi:10.1016/j.ction

trends towards automation of electroanalytical mea-impose increased demands on all procedures usedesearchprocess. Theprocedures should be automatic asssible, and capable of being performed under computerst of the current effort has concentrated on automatingation for electroanalytical experiments [13]. However,ripe to develop automatic procedures for theoretical4], computer simulation [5,6], and electrochemical datawell. Such developments would enable, for example,of computer-operated intelligent sensors capable of

classication of the experimental observations, black-es for in-situ data analysis, or a new generation oflving Environments [7,8], i.e., advanced software sup-researchers in their intellectual activities such as

ding, model-solving, hypothesis testing, etc., within the

dence address: Institute of Physical Chemistry of the Polish Academypartment of Complex Systems and Chemical Processing of Informa-ominajek 8, 30-239 Cracow, Poland. Tel.: +48 12 639 52 12;5 19 23.ress: nbbienia@cyf-kr.edu.pl.//www.cyf-kr.edu.pl/nbbienia.

eldof theoretical andcomputational electrochemistry [9]. Inorderto contribute to the progress in this area, in the previous works ofthe present author [1012] an adaptive method for solving rstkind Abel integral equations (IEs) has been developed. The methodcan be used for the simulation of transient electroanalytical exper-iments [13] such as, in particular, cyclic voltammetry (CV) [14].The description of the transient experiments in terms of the IEs,resulting from the Laplace transform of kinetic partial differentialequations (PDEs), is a classical theoretical methodology [1517].Although this approach is limited to linear PDEs, and thereforeis not very general, it often proves useful and efcient, when itis applicable. Based on a number of example kinetic models itwas shown [12] that the adaptive Huber method [10,11] enabledcomputation of theoretical CV current functions with a prescribedaccuracy,without anyaprioriknowledgeof the temporal behaviourof the functions, and corresponding optimum discrete time gridsneeded for calculations. Numerical methods possessing such fea-tures are often called automatic, and we adopt this terminologyin the present paper, when writing about automatic simulations.Any broader understanding of the automatism of the simulations,possibly preferred by more demanding readers, is not addressedhere. Furthermore, the adaptive solution of the IEs was found moreefcient than the direct solution of kinetic initial boundary valueproblems by the patch-adaptive nite-differencemethod [18]. Thisis partially caused by the fact that spatio-temporal concentration

see front matter 2009 Elsevier Ltd. All rights reserved.electacta.2009.09.022. Bieniasza,b,

ysical Chemistry of the Polish Academy of Sciences, Department of Complex Systems anajek 8, 30-239 Cracow, Polandrsity of Technology, ul. Warszawska 24, 31-155 Cracow, Poland

e i n f o

y 2009vised form 1 September 2009tember 2009e 15 September 2009

l electrochemistryomationetry

a b s t r a c t

Considering modern trends of laboramethods for automatic simulation ofa topical issue. One of the classical simrelevant integral equations. In formeHuber method serving for this purpoexamples of rst kind Abel integral ekind Volterra integral equations withbetween the unknowns and their inteelectrochemical simulations, is tested/ locate /e lec tac ta

y the adaptive Huber methodequations

mical Processing of Information,

automation in electroanalysis, the development of adaptiveochemical transient techniques such as cyclic voltammetry ison approaches relies on formulating, and solving numerically,rk of the present author an adaptive variant of the populars been proposed, and successfully tested on electrochemicalons (IEs). The method has been recently extended to secondkly singular kernels and linear and non-linear dependences. In the present work the validity of the extended method, forrepresentative examples of such equations, occurring in the

722 L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728

proles need not be computed in the integral equation method,so that the related computational cost is saved. In the most recentwork [19] the adaptiveHubermethod [10,11] has been extended tosecond kind Volterra IEs with weakly singular kernels. In contrastto the clascal/numericthan one intionships bebeen alloweThe purposextended mthe IEs, occof the IEs oin kinetic mcharge tranimportant cmodels repfor reversibare, in turnear reactionnon-linearoccur whenheterogenecients or rethe IEs arishomogeneouncompens

2. The met

As the din detail inWe provideelectrochem

F(t, (t), Y1

where (t)dimensionl[15]), Yi(t) (

Yi(t) = t

0

K

with kernelar, and F()focus here oKi(t, ) = (tKi(t, ) = 1. Ain electrochthe framewies. The adacontinuousEq. (1) mustransformaa sum of sinnon-singulais calculate

The adasation [24]controllingwise lineargrid of node. . ., N). Appdeterminedfunction (

replaced by quadratures yi,n:

yi,n =n

k=1

tkt

Ki(tn, )[k1 +

k k1hk

( tk1)]

d

nk=1

(

uadr

= tkQ

= Qi,

,k =

se cont fo,19])e deonenera

n, y1,

ne unethode 0e ap

ons

, 1/

1, y1

two)/21/201. Nothe papp

y erred ine excalcufor kls [0aredthey

acingr n>the i

n, )

0 anerror

=

timat, by nsical IEs of this type, considered in the mathemati-al literature, a dependence of the equations on moretegral of the unknown function, and non-linear rela-tween the unknown function and/or its integrals, haved, tomeet requirements of electroanalytical chemistry.

e of the present study is to investigate the validity of theethod for the solution of typical examples of this sort ofurring in the CV theory. The simultaneous dependencen the current function and its integral(s) occurs mostlyodels involving totally irreversible or quasi-reversiblesfer reactions [15]. This is a much larger and morelass of models, compared to the previously consideredresented by the rst kind Abel IEs that originate mostlyle charge transfer reactions. Non-linear dependencies, characteristic of kinetic models represented by lin-diffusion partial differential equations coupled withboundary conditions. The latter boundary conditionsthe models involve charge transfer reactions or other

ous reactions having non-unity stoichiometric coef-action orders [2022]. Other kinds of non-linearities ine from the assumption of quasi-steady state for someus reactiondiffusion systems, or from the effects ofated ohmic resistance on CV curves [23].

hod

erivations of the adaptive method have been presenteda separate article [19], they will not be repeated here.nal equations only. The method can be applied toical IEs that take the form:

(t), . . . , YI(t)) = 0 (1)

is the unknown CV current function dependent oness time t (often denoted by at or bt in the literaturefor i=1, . . ., I) are integrals

i(t, )()d (2)

ls Ki(t, ) among which at least one is weakly singu-is generally a non-linear function of its arguments. Wen three kernels often encountered in electrochemistry:)1/2, Ki(t, ) = exp[(t )](t )1/2 with >0, andn identication of other types of kernels that can occuremical kinetics, and incorporation of such kernels intoork of the present method, necessitates further stud-ptive method requires, among other things, that (t) beat t=0. In cases when this assumption is not satised,t be transformed to an appropriately regular form. Thetion most likely involves a decomposition of (t) intogular and non-singular components, out of which ther one is determined numerically, and the singular one

d analytically or ignored.ptive method [19] combines the Huber-type discreti-with an automatic integration step selection based onlocal error estimates. Thus, (t) is replaced by a piece-spline function, spanned on the generally non-uniforms tn (n=0, 1, . . ., N) with step-sizes hn = tn tn1 (n=1,roximate nodal values n of the current function are, one after another, in the following way. For any n>1t) in Eq. (1) is replaced by n, and integrals Yi(t) are

=

with q

Ri,n,l,k

and

Si,n,l,k

where

Qi,m,n,l

All the(releva[10,11must bof onlythe (ge

F(tn,

with oton mbecauspose, wequati

F(t1/2

F(t1,

with(t0 + t1Ri,1/2,0,Si,1,0,1t1/2 in

The(tn) bobtainwith thwhileues kintervaand 1so thatof repli,n fotion in

tntn1

Ki(t

Errorslation

(t1/2)

The esof i,nk1

Ri,n,k1,kk1 + Si,n,k1,kk) (3)

ature coefcients

i,0,n,l,k Qi,1,n,l,ktk tl

(4)

1,n,l,k tlQi,0,n,l,ktk tl

, (5)

tktl

Ki(tn, )md. (6)

efcients are determined analytically for a given kernelrmulae for the kernels listed above have been provided. Since for n>1 all previousk values for k=0, 1, . . ., n1termined earlier, yi,n is, according to Eq. (3), a functionunknown n, i.e., yi,n = yi,n(n). Consequently, we solvelly non-linear) algebraic equation

n(n), . . . , yI,n(n)) = 0 (7)known n. This is accomplished by the standard New-. The procedure is more complicated at n=0 and 1,and 1 are determined simultaneously. For this pur-ply the Newton method to the system of two algebraic

2(0, 1), y1,1/2(0, 1), . . . , yI,1/2(0, 1)) = 0,1(0, 1), . . . , yI,1(0, 1)) = 0

}(8)

unknowns 0 and 1, in which t1/2 =with t0 = 0, 1/2(0,1) = (0 +1)/2, yi,1/2(0, 1) =+ Si,1/2,0,1/2(0 +1)/2, and yi,1(0, 1) =Ri,1,0,10 +n-integer indices in the coefcients mean that we takelace of tn or tk in Eqs. (4)(6).roximate solutions n differ from the true solutionsors n =(tn)n. Estimates est(n) of these errors arethe followingway. First, the errors are regarded as local,ception of the rst two errors 0 and 1. This means thatlating the error of n for n>1 all previous nodal val-= 0, 1, . . ., n1, and the current function integrals over, tn1], are considered as exact. The rst two errors 0eterminedsimultaneouslywithout suchanassumption,are global errors. Errors n for n>1 are a consequenceintegrals (2) by quadratures (3). The quadrature errors

0 (including the case of n=1/2, which denotes integra-nterval [0, t1/2]) are dened by:

()d=[Ri,n,n1,n(tn1)+Si,n,n1,n(tn)]+i,n, (9)

d 1 are additionally inuenced by the linear interpo-1/2 at t1/2:

(t0) + (t1)2

+ 1/2 (10)

es est(i,n) of i,n are obtained from Taylor expansionseglecting second and higher expansion terms, and by

L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728 723

replacing second derivatives of (t), occurring in the expansions,by standard three-point central nite-difference quotients [25]:

est(i,n) =

Wi,1/2(0 21 + 2) forn = 1/2

where n =obtained asapproximatThe interpo

est(1/2) =

Coefcientsdependent

Vi,n =12

h2n

+Qi,

Wi,1/2 =12

h

The CV potalways forcthen the se(11) for n>

est(i,n) =where n+1means thattn1. Linearto the linea

Jn est(n) =

[J0,0 J0,1J1,0 J1,1

]

=

F,1

Ii=1

Symbols F,to (t) andas a functioF() in Eq. (8are evaluatthe case of(in the case

After dealgorithm o[12], in ordecriteria are

max{est(est(n)

est(n)

where tol is a prescribed absolute error tolerance, and is a heuris-tic safety factor, chosen to account for the discrepancies betweenthe true and estimated errors, which are anticipated in cases whenthe current function is not differentiable at t=0 or at t= ts [11,12].

lue art isexpeked,ent

o bedditoutpovidthepoly

tivesointusedn vamplxperi

mpl

demampf theve m

amp

curreRn, taler etionaies Ospectraigs uby t

t

0 (t

l u isionlg pon, ansufcmboial sw

={

) can

()

)1/

ical(t) i

much, thee negVi,1(0 21 + 2) forn = 1

Vi,n

[2

n(1 + n)n2 2n n1 +

2(1 + n)

n

]forn > 1

(11)

hn1/hn is the local grid ratio, 2 is a provisory solutionsuming t2 =2h1. The provisory solution serves only foring the second derivative at t1, and is later discarded.lation error 1/2 is estimated analogously as

18(0 21 + 2) (12)

Vi,n and Wi,1/2 in Eq. (11) are additional kernel-method coefcients dened by [19]:

[(t2n1 + hntn1)Qi,0,n,n1,n (2tn1 + hn)Qi,1,n,n1,n

2,n,n1,n] forn 1, (13)

21 (Qi,2,1/2,0,1/2 h1Qi,1,1/2,0,1/2). (14)

ential sweep switching time ts (cf. Eq. (22) below) ised to coincide with a grid node. If ts = tn1 for some n,cond derivative d2(ts)/dt2 does not exist, and formula1 is replaced by

Vi,n(n1 2n + n+1) (15)is aprovisory solutionobtainedassuminghn+1 =hn. Thisthe second derivative is evaluated at tn rather than atization of Eqs. (7) and (8) around (tn) and Yi(tn) leadsr algebraic equations for est(n):

I

i=1FYi,nest(i,n) forn > 1 (16)

[est(0)est(1)

]

/2est(1/2) +I

i=1FYi,1/2est(i,1/2)

FYi,1est(i,1)

forn = 0and1.

(17)

n and FYi,n denote partial derivatives of F() with respectYi(t), Jn is the total derivative of F() in Eq. (7), regardedn of n, and J0,0, J0,1, J1,0 and J1,1 are partial derivatives of), regarded as functions of 0 and 1. These derivativesed at their discrete arguments corresponding to: tn (inF,n, FYi,n and Jn), t1/2 (in the case of J0,0 and J0,1), and t1of J1,0 and J1,1).termining est(n) for n=0, 1, . . ., the control theoreticf Gustafsson [26] is applied identically as in Bieniaszr to adjust the step-sizes so that the requested accuracymet. The criteria are:

0) , est(1)} tol forn = 0and1 (18)

tol forn > 1and tn1 /= ts (19)

tol forn > 1and tn1 = ts (20)

The vah1 =hstvalue,is checdependsteps t

In adensealso prthat ofmitianderivathree-pcan befunctiofor exawith e

3. Exa

Weve exbasis oadapti

3.1. Ex

CVO+e

reactioby Kohproporof specux ofmore ssionlesal. [22][1

Symbodimenstartinreactiotakenble. Sypotent

S(t, ts)

Eq. (21

t0

(t

Numerwhen (t) isto zerobecom=10 has been used in this work. On the rst time step,initially attempted, where hstart is a predened step

cted to be optimal for a given IE, and the criterion (18)possibly resulting in h1 modications. A maximum, IE-limit hmax for the steps helps to avoid excessively largetaken.ion to determining nodal solutions n, the so-calledut [27] of the inter-nodal current function values is

ed. The accuracy of the dense output is comparable ton values. This is accomplished by the third order Her-nomial interpolation of the nodal n values, with rstof the current function (at grid nodes) approximated bynite-difference formulae. The dense output formulaeto additionally obtain approximate theoretical currentlues on any arbitrarily chosen time or potential grids,e on uniform grids that can be useful for comparisonsmental transients.

es

onstrate the operation of the adaptive method usingles of Eq. (1). The examples are chosen mostly on their mathematical properties enabling the testing of theethod in the most transparent way possible.

le 1

nt function for the fully irreversible electrode reactionhaving an arbitrary reaction order . The IE for such aking place at a planar macro-electrode, was formulatedt al. [22]. Taking into account that the reaction rate isl to cO(0,t) , where cO(0,t) is the surface concentration, they dened the current function as the dimensionlessies O at the electrode, raised to the power of 1/. It ishtforward, however, to simply dene(t) as the dimen-x of O, which allows one to replace Eq. (12) in Kohler ethe following IE:

()

)1/2d

]

exp[u S(t, ts)](t) = 0. (21)

n Eq. (21), and in further examples below, denotes aess parameter dependent on the difference between thetential and the formal potential of the charge transferd on other model parameters. The value of u must beiently large for the effect of u on (t) to be negligi-

l ts denotes the dimensionless duration of the forwardeep, and

t for t ts2ts t for t > ts . (22)

be written in the equivalent alternative form:

2d + exp

[u S(t, ts)

](t)1/ 1 = 0. (23)

experimentation revealed that form (21) is preferables close to zero, whereas form (23) performs better when

different from zero. This is because when (t) is closen during the Newton iterations (t) may occasionallyative. In such cases,(t)1/ in Eq. (23) is not computable

724 L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728

Fig. 1. Cyclic voltammetric current functions simulated adaptively in the interval t [0, 30], with model parameters u=8 and ts = 15, and method parameters tol=104,hstart = 0.01 and hmax =1. The various symbols denote discrete current function values, obtained at the nodes of the temporal grid, according to the following specication:(a) example 1 with =0.5 ( ), 1 (), 1.5 (), 2 (), 2.5 () and 3 (); (b) example 2 with =0.1 ( ), 0.5 () and 1 (); (c) example 3 ( ); (d) example 4 with =50 andp=1 ( ), 2 () and 3 (); and (e) example 5 with H=0.1 ( ), 5 () and 10 (). Solid lines denote dense output by third order Hermitian interpolation.

L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728 725

for certain , for example for =2. The occurrence of unaccept-able argument values causes domain errors of the C functionpow( ) used in calculations. At the same time, however, the term1

t0(()/(t )1/2)d in Eq. (21) is far from zero, so that there

is a small pmay occasiofor other exis far fromfor any , wmay becomused for tsite case. Wintegral is ntion for (tclassical IEothers by Nnite seriesknown [15,

(t) = 1/

where x=be truncateof the succezero effectwhen the cadouble variaseries (24) ibe accompltu not larcomputer apoint errors

3.2. Examp

CV curnism involand a pseR+AO+cients of O t

0

exp[(t

where >0reaction. Thintegral invcients [10(24) exists f

(t) = 1/

If is sufc(26) increasthe truncatmade in thisummationently not la

(t u)max =

The validity

3.3. Example 3

CV current function for the ECmechanism involving a reversiblecharge transfer and a fast irreversible follow-up homogeneous

zatiouns

t

0 (t

lytic1) anl, are

amp

modeelecRis the:

(

t p denctrodnstant kecal s

amp

curreRt fun:

tot(

)1

H isto bee of ictualbe c

becaally jand

uatioimplptionmiceu. Oversid thsonseactiineistenrentwhe

) thu

step(

)1robability that during the Newton iterations the termnally become negative causing similar domain errorsponents, such as for example =1/2. When in turn (t)zero, then (t)1/ in Eq. (23) can be safely computedhereas the term 1

t0(()/(t )1/2)d in Eq. (21)

e negative, causing problems. Therefore, Eq. (21) was[0, ts/2] (3ts/2,), and Eq. (23) was used in the oppo-henever /= 1, the relationship between (t) and itson-linear in Eqs. (21) and (23), and no analytical solu-) is available. For =1 Eqs. (21) and (23) reduce to thefor the rst-order electrode reaction, reported amongicholson and Shain [15], for which the analytical in-solution for the forward part of the voltammogram is28]:

2j=1

(1)j+1[(j 1)!]1/2xj, (24)

1/2 exp(tu). In practical calculations, the series mustdafter anitenumberof terms. In thiswork theadditionssive terms was continued as long as this had a non-

on the oating point representation of the partial sum,lculationsweremade in extended precision (C++ longbles having 80bits and18digit precision). Although thes formally convergent for any t, such a summation canished (on a contemporary personal computer) only forger than about unity, due to the nite precision of therithmetic, and related numerical instability and oating.

le 2

rent function for the catalytic reaction mecha-ving irreversible charge transfer reaction O+e Rudo-rst-order irreversible homogeneous reactionB. Assuming planar electrodes and equal diffusion coef-and R, the current function obeys the linear IE [15]:

(t )])1/2

()d + exp[u S(t, ts)](t) 1 = 0 (25)

is the dimensionless rate constant of the homogeneousus, Eq. (25) is similar to Eq. (23) with =1, but the

olves a different kernel requiringdifferentmethod coef-,19]. An analytical innite series solution similar to Eq.or the forward part of the CV wave [15]:

2j=1

(1)j+1[

j1i=1

( + i)]1/2

xj. (26)

iently large, the presence of a large denominator in Eq.es somewhat the range of t in which the summation ofed series (26) can be performed on a computer. Testss work suggest that the largest tu value for which thegives a result with the absolute error modulus appar-rger than 107 is{

1 for 102log 1 for > 102 . (27)

of Eq. (27) was checked up to =1012.

dimerifor the[23]:[1

An anaEqs. (2integra

3.4. Ex

CVof thesteps:modelby (t)

p

t0 (

wherethe elerate codiffereanalyti

3.5. Ex

CVO+e

CurrenIE [23] t

0

(t whereneedsbecaustrol. Acannotarisesphysicboth uthe eqwhichassumnite ohofnitor irrerent anthe reasuch rdeterminconsthe curat t=0step(t t

0

(t n reaction: O+eR, 2RZ. Assuming a steady statetable intermediate R, the current function obeys the IE

()

)1/2d

]3 exp{3[u S(t, ts)]}(t)2 = 0. (28)

al solution does not exist. A novel feature, compared tod (23), is that the dependences of the IE, on (t) and itsboth non-linear.

le 4

l of Aoki and Kato [29], for the deposition/precipitationtrochemical product, proceeding through reversibleRad, RadOad + e. The essential component of theIE for the dimensionless ux of species R, here denoted

)

)1/2d +

1 + t0

()d

1 + exp[S(t, ts) u] +(t)

1 = 0, (29)

otes the ratio of the amount of Rad to a supply of R intoe by diffusion, and is the dimensionless desorption

nt. The IE is linear, but it involves two integrals withrnels, which presents a complication for simulation. Anolution is not available.

le 5

nt function for the reversible charge transfer reactionin the presence of uncompensated ohmic resistance.ction tot(t), representing the total current, obeys the

)/2

d 11 + exp[u S(t, ts)] exp[Htot(t)] = 0, (30)

the dimensionless uncompensated resistance. Eq. (30)reformulated before applying the adaptive method,

ts self-inconsistency at t=0, which precludes error con-ly, the literature numerical solutions [23] of Eq. (30)orrect close to t=0, for this reason. The inconsistencyuse the reversibility of the charge transfer contradictsustied nite u and nite ohmic drop. Assuming thattot(0) are nite in the exponential terms in Eq. (30),n predicts that the integral must be non-zero for t=0,ies tot(t) t1/2 when t0, in disagreement with theof nite tot(0). Innite tot(0) implies, in turn, an in-

potential drop, which is in conict with the assumptionf course, thisproblemdoesnot arise forquasi-reversibleble charge transfer reactions, for which both the cur-e potential drop is always nite. This may be one ofwhy the effect of the uncompensated resistance for

ons has been more frequently modelled. In order tothe CV response without the interference of the initialcy,we substitutetot(t) =step(t) +(t),wherestep(t) isfunction component associated with the potential stepreas (t) is the actual CV component. The components obeys the IE:

)/2

d 11 + exp[u + Hstep(t)] = 0, (31)

726 L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728

which encapsulates the initial inconsistency, and (t) obeys the IEresulting from the subtraction of Eqs. (30) and (31): t

0

()

(t )1/2d

{1

1 + exp[u + Hstep(t)] exp[H(t) S(t, ts)]

11 + exp[u + Hstep(t)]

}= 0. (32)

Eq. (32) is self-consistent at t=0, and can be solved adaptively, pro-vided thatwe have a reasonable approximation for step(t). Findingone may seem hopeless, at rst sight, because of the inconsistencystill present in Eq. (31). However, the theory of the uncompen-sated resistance effect on potential step transients [30,31] revealsthat largely independently of the kinetic assumptions that areimportant close to t=0, for larger times the currents rather quicklyapproach the Cottrellian currents. Therefore, by omitting the ohmicdrop in Eq. (31), we obtain that for sufciently large time

step(t) 1[1 + exp(u)]1t1/2. (33)

Setting Eq. (33) into Eq. (32) yields a good approximation, becauseirrespectively of the actual initial behaviour of step(t), the twoterms in braces in Eq. (32) cancel during a longer time, when (t)and S(t, ts) do not depart meaningfully from zero. In other words,we do not need to know step(t) very exactly, in order to obtaincorrect (t) from Eq. (32).

4. Computational details

Computational experiments were performed on an IBM-compatibleoperating acodewaswblevariableIEEE 754 stusing BorlanWindows X

5. Results and discussion

Fig. 1 presents typical plots of the simulated (t) functions.As can be seen, the grid nodes automatically concentrate in thet regions where an increased curvature of the functions occurs, forexample in the vicinity of the peaks. This observation qualitativelyconrms the correctness of the adaptive grid selection. A directquantitative verication of the adaptive results is possible only par-tially, specically in example 1 with =1, and in example 2, whereanalytical reference solutions (24) and (26) can be computed. Fig. 2compares the absolute true global errors obtained in these cases,with the a posteriori local error estimates. The comparison revealsthat the agreement between the two kinds of errors is fairly good inexample 2,where it is similar to the previously studied examples ofthe rst kind Abel IEs [1012]. The agreement is worse in example1 with =1, where the true errors are even about ve times largerthan the estimated ones. Similar observations have beenmade for anumber of non-electrochemical examples of Eq. (1) [19]. Therefore,we conclude that in the cases when Eq. (1) depends simultane-ously on (t) and its integral(s), the local error estimates tend to beover-optimistic approximations to the trueglobal errors, anddonotfully guarantee achieving the true errors at the level of the prede-ned tolerance parameter tol. The conclusion refers to the kernelsKi(t, ) = (t )1/2,Ki(t, ) = 1, andKi(t, ) = exp[ (t )](t )1/2with not greater than a certain limiting value. The situation is dif-ferent for the latter kernel, when is larger. With increasing thetrue errors decrease considerably, and for sufciently large maybecome much smaller than tol. The phenomenon is illustrated byFig. 3. Similar error reduction has already been observed for therst kind Abel equations involving such kernels [12].

In an ideal adaptive method, the true errors achieved should beequal to the parameter tol [32], or at least proportional to tol, with

r coele 1d noseried cot ava

Fig. 2. Estima t funcexamples: 1 w densereference curr wheremethod parampersonal computer having an Intel PentiumDprocessort 3GHz, and a 2GB operational memory. The numericalritten in C++ usingmostly extended precision (long dou-shaving80bit and18digitprecision, compliantwith theandard), and compiled as a 32-bit console application,d C++ Builder 6.0 compiler. The codewas run underMSP Professional. Further details are as in Bieniasz [19].

a lineaexampthe griof theues, anare no

tes est(n) of the absolute local errors of the nodal values of the simulated currenith =1 (a); and 2 with =1 (b). Solid lines represent true absolute errors of theent function values are determined from Eqs. (24) and (26), within the range of t,eters are as in Fig. 1.fcient close to unity. This property was examined inwith =1, and in example 2. The largest true errors atdes located in the t range of the practical summabilitys (24) and (26) were computed for a number of tol val-mpared with tol. For other examples analytic solutionsilable, so that the true errors could not be determined.

tions (), compared with the true absolute errors (), obtained foroutput by the third order Hermitian interpolation. Highly accuratethe summation of the series is feasible according to Eq. (27). Other

L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728 727

Fig. 3. Effect oabsolute errorwithin the -daccording to E

We therefothe followinperformed,ward peakmodel paraerence valunode positiotive simulatand the simapproximatgrid nodesobtained inically detererrors of thof the aboving tol are sis obtainederror measu(with H=5)to linear, butol values inare greaterdences of tlinearity,mFig. 4 are leAbel IEs [10possible torange wherFig. 4 and si

Table 1Reference valumined in calcu

Example

1345

ffect of the error tolerance parameter tol on the error measure ERR of thely simulated current functions in examples: 1 with =1 ( ); 1 with =2ith =1 (); 3 (); 4 with =50 and p=1 (); and 5 with H=5 (). Ins 1 with =1, and 2 with =1, the simulation is performed only for t [0,tity ERR is then equal to the largest modulus of the absolute nodal errord in this interval. In other examples, the simulation is performed for t [0,ts = 15, and ERR is the modulus of the difference between the dense output

function value at tfp, and the reference value fp from Table 1.

t that for a given tol, the absolute values of the absolute trueare n5 revordeerrorhis ruf parameter in example 2, on the largest absolute values of the trues of the simulated nodal values of the current functions, observedependent range of t, where the summation of series (26) is feasible,q. (27). Other method parameters are as in Fig. 1.

re resorted to studying convergence plots obtained ing way. First, adaptive simulations with tol=107 wereto obtain the most accurate reference values of the for-values fp of the current functions, for a few selectedmeter values in the various examples. The resulting ref-es are summarised in Table 1, together with the gridns tfp at which the maxima were observed. Next, adap-ionswith a number of larger tol valueswere performed,ulated values of(tfp)were comparedwithfp to obtaine measures of the true errors. Since for tol /= 107 thedo not generally coincide with tfp, the error measuresthis way depend not only on the errors of the numer-mined nodal current function values, but also on thee dense output by Hermitian interpolation. The results

Fig. 4. Eadaptive(); 2 wexample9]. Quanobserve30], withcurrent

suggeserrors

Fig.by oneof thefrom te examination of the error convergence with decreas-hown in Fig. 4. Seemingly, the most satisfactory resultfor example 4 (with p=1 and =50), where the trueres are close to tol. For examples: 2 (with =1) and 5, the dependences of the error measures on tol are closet the differences between the true error measures andcrease with decreasing tol, and the true error measuresthan tol. Finally, in the remaining examples the depen-he true error measures on tol exhibit deviations fromost likely fortuitous in nature. Although the results fromss satisfactory than analogous results for the rst kind12], it is clear that by appropriately decreasing tol it isreduce the true errors as much as is desired, within thee the interference of machine errors is absent. Based onmilar results for non-electrochemical examples [19]we

es of the current function forward peak positions and heights, deter-lations with tol=107.

Parameters tfp fp

=2 8.9457308926 0.21758045477 8.5188273954 0.29727818820p=1, =50 9.6917202370 0.24264775157H=5 10.649857530 0.23477237874

Fig. 5. Dependfunctions, onparameters arto tol=103, 5106, 5107plots).ot very likely to exceed 100 tol, in the worst case.eals that a change of the number N of integration stepsr of magnitude corresponds, on average, to the changes by two orders of magnitude (with some departuresle in individual examples, reecting the irregularitiesences of the error measure ERR, of the adaptively simulated currentthe total number N of integration steps needed. Notation and modele as in Fig. 4. The successive data points (from left to right) correspond104, 2104, 104, 5105, 2105, 105, 5106, 2106,, 2107, 107 (the last three values are not represented in some

728 L.K. Bieniasz / Electrochimica Acta 55 (2010) 721728

Fig. 6. Dependfunctions, on tages over 500in Fig. 4, and p

already discthe practicais inverselythe method[19]. Finallymethod. Thinversely pare analogoelectrochemcomparableshorter thanfor kinetic P

6. Conclus

The resuHubermethandefcienring in the tis similar torst kind A

simultaneously on (t) and its integral(s), the true errors of thesimulated current functions may deviate from the prescribed errortolerance parameter tol. Judging from the examples considered, inthe worst case the true errors may reach even 100 tol. Therefore,tol must be chosen more conservatively in cases when achievinga target accuracy is a crucial issue. Nevertheless, the method isa useful candidate for automatic IE solvers, needed for building anew generation of Problem Solving Environments for electrochem-istry, and for thewidelyunderstoodautomationof electroanalyticalinvestigations.

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ussed in connectionwith Fig. 4). This result implies thatl accuracy order is close to 2, since an average step sizeproportional to N. Theoretically, the accuracy order ofis close to 2, but not necessarily equal to 2, for Eq. (1), Fig. 6 presents efciency plots for the adaptive Hubereplots reveal that the computational time is, onaverage,roportional to the error achieved. These observationsus to the characteristics of the adaptive solution of theical rst kind Abel IEs [12]. Simulation times are also, and vary between 0.0007 and 0.9 s, so that they aretypical simulation times of thepatch-adaptivemethodDEs [18], needed for a comparable accuracy.

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solCom

[8] E.N[9] L.K

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[27] E. HairerNonstiff P

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Automatic simulation of cyclic voltammograms by the adaptive Huber method for weakly singular second kind Volterra integral equationsIntroductionThe methodExamplesExample 1Example 2Example 3Example 4Example 5

Computational detailsResults and discussionConclusionsReferences