Numerical solution of nonstationary problems for a system of nernst ...

ISSN 20700482, Mathematical Models and Computer Simulations, 2013, Vol. 5, No. 3, pp. 229–243. © Pleiades Publishing, Ltd., 2013. Original Russian Text © P.N. Vabishchevich, O.P. Iliev, 2012, published in Matematicheskoe Modelirovanie, 2012, Vol. 24, No. 10, pp. 133–148.

Numerical Solution of NonStationary Problems for a System of Nernst–Planck Equations P. N. Vabishchevich and O. P. Iliev a

Nuclear Safety Institute, Russian Academy of Sciences, Moscow, Russia b Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany email: [email protected] Received October 13, 2011

Abstract—The mathematical model of electricity and mass transfer is based on the equations for charged particles (ions and cations) in the electrolyte, i.e., the Nernst–Planck equations. These equa tions are supplemented by the equation for the electric field and the motion equations for the electro lyte as a continuum. The paper mostly focuses on the construction of the time approximation for the approximate solution of nonstationary problems. The system of Nernst–Planck equations is charac terized by a quadratic nonlinearity. Special linearization schemes are proposed for taking this into account. Computational algorithms are studied on a model problem for the binary electrolyte (two kinds of charged particles). Keywords: electricity and mass transfer, Nernst–Planck equations, difference scheme, quadratic non linearity DOI: 10.1134/S2070048213030125

1. INTRODUCTION We write out a system of equations describing the charge transfer in the electrolyte. In this case, the fundamental issue is the allowance for the influence of the transfer of mass on electric fields. We introduce the basic notation for the description of electrochemical systems [1]. The considered solution consists of unionized solvent, electrolyte in the form of ions, and uncharged components. Let ci be the concentration of the charged component i, z i be the charge number of the component i, and the total number of charged particles be equal to p, i.e., i = 1, 2,..., p. The movement of the components (ions and cations) is described by the flux vector Ni. The material balance (mass conservation law) for an individual component can be written as the equation

∂ci (1) = −∇N i + Ri , ∂t where Ri is the source term (bulk chemical reaction) and t is time. When the reactions only proceed on the electrode surface Ri = 0. The movement of ions is due to the motion of the entire melt at a rate of v (free convection). For an inhomogeneous medium the movement of ions is also due to diffusion. The third mechanism is related to the motion of charged particles (ions) in the electric field. In view of this, for the flow of an individual component, we have N i = −z i ui Fci∇ϕ − Di∇ ci + ci v.

(2)

Here, ui is the component’s mobility, F is the Faraday constant, ϕ is the electrical potential, and Di is the diffusion coefficient of the component. The motion of charged components leads to the generation of an electric current in the electrolyte solution. For the current density vector j we have Faraday’s law p

j=F

∑z N . i

i =1

229

i

(3)

230

VABISHCHEVICH, ILIEV

The electric field due to the volume distribution of the charged components is described by the equation p

∇(ε∇ϕ) = −F

∑z c ,

(4)

i i

i =1

where ε is the dielectric permittivity. Taking into account the low value of the dielectric permittivity, for practically important cases we can use the approximation of electroneutrality when instead of (4) we employ the ratio p

∑z c

i i

= 0.

(5)

i =1

In modeling specific electrochemical processes the system of the Nernst–Planck equations (1)–(4) is supplemented by the appropriate boundary and initial conditions. The main peculiarities of these pro cesses are often manifested exactly at the electrolyteelectrode interface. Problems arising in connection with the adequate description of electrochemical processes on electrodes are extremely complex. This is due, in particular, to the multicomponent electrolyte composition and multiplicity of chemical reactions. The focus is mainly on the assignment of specific dependences of the surface density of the electric current on the electrical potential on the electrode and in the electrolyte and on the allowance for the electrolyte composition. This typical situation is associated with nonlinear boundary conditions for potential ϕ and concentration ci , i = 1,2,..., p. The Butler–Volmer conditions [1, 2] are a good example of such boundary conditions. In this paper, the problems of boundary conditions are not discussed, as we restrict ourselves to problems with simple Dirichlet and Neumann boundary conditions for the potential and concentra tions. Currently, only a few attempts have been made at a numerical simulation of electrochemical systems based on the numerical solution of boundary value problems for a general multidimensional system of the Nernst–Planck equations (1)–(4) (see [3, 4]). Computationally, we should separately mention study [5] mainly focused on the construction of a splitting scheme without allowance for hydrodynamic effects. Issues associated with the numerical solution of nonstationary boundaryvalue problems for the Nernst– Planck equations without allowance for convective transport are discussed in [6]. Under the finiteele ment space approximation the twolayer purely implicit time scheme is employed. We also consider the scheme with a special approximation of nonlinear coefficients in terms that describe in (2) the electric drift of charged components. The structure of the present paper is as follows. In Section 2 a nonstationary boundaryvalue problem for the system of Nernst–Planck equations (1)–(4) is considered. Purely implicit and linearized time schemes are constructed. Standard finiteelement or finitedifference space approximations are used. The testing of time schemes is based on a model twodimension problem in a rectangle for the system of the Nernst–Planck equations (1)–(4) with two charged components (Section.4). In the last part of the paper, the results obtained by the numerical solution of the test problem using different time schemes are pre sented. 2. PROBLEM STATEMENT Let Ω be a limited twodimensional (m = 2) or threedimensional (m = 3) domain x = (x1, x2,..., x m ). We will consider the boundaryvalue problem for the system of the Nernst–Planck equations in the absence of volume reactions for electrically charged components and movement of the electrolyte. For the concentrations from (1) and (2), we obtain

∂ ci = ∇(Di∇ci + z i ui Fci∇ϕ), i = 1,2,..., p, x ∈ Ω, 0 < t ≤ T . ∂t The electric field is described by the equation

(6)

p

∇(ε∇ϕ) = −F

∑z c , i i

x ∈ Ω, 0 < t ≤ T .

(7)

i =1

The system of equations (6) and (7) for ϕ(x, t ), ci (x, t ), i = 1,2,..., p, is supplemented with the appropri ate initial and boundary conditions. The initial state is determined by the given concentrations

ci (x,0) = ci0(x), i = 1,2,..., p, x ∈ Ω. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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We restrict ourselves to the simplest boundary conditions for the system of equations (6), (7). On one part of the boundary ∂Ω = Γ1 ∪ Γ 2 the potential is specified, on the other part the condition of electrical insulation is assigned, ∂ϕ (9) ϕ(x, t) = g(x, t ), x ∈ Γ1, ε (x, t ) = 0, x ∈ Γ 2, 0 < t ≤ T , ∂ν where ν is the normal to the boundary. We assume that the boundaries are impermeable for the compo nents, i.e., ∂c ∂ϕ (10) Di i + z i ui Fci (x, t ) = 0, i = 1,2,..., p, x ∈ ∂Ω, 0 < t ≤ T . ∂ν ∂ν These boundary conditions correspond to the problem of the charged components’ redistribution under the influence of the applied voltage g(x, t), x ∈ Γ1 on a part of the boundary. Among the properties of the solution of problem (6)–(10), we note the law of mass conservation of individual components. From (6), (8), and (9) it immediately follows that

(

)

∫

∫

M i ≡ ci (x, t)dx = ci0(x)dx, i = 1,2,..., p, 0 ≤ t ≤ T . Ω

(11)

Ω

It would be natural to require the fulfillment of these conservation laws at the discrete level, i.e., the conservative approximations [7]. We can also mention the property of the solutions of the considered problem associated with the nonnegativity of ci (x, t ), i = 1,2,..., p. Under the constraints

ci0(x) ≥ 0, i = 1,2,..., p, x ∈ Ω, we have

ci (x, t ) ≥ 0, i = 1,2,..., p, x ∈ Ω for all 0 < t ≤ T . For Eq. (6) nonnegative solutions are ensured by the maximum principle [8]. At a discrete level, this property is associated with monotonic approximations [7]. We assume for simplicity that Di , i = 1,2,..., p, and ε are constant. In this case the boundaryvalue problem (6)−(10) is characterized by a quadratic nonlinearity—the terms in Eqs. (6), which describe the drift in the electric field. 3. TIME DIFFERENCE SCHEMES For the approximate solution of problem (6)–(10), we use ordinary twolevel schemes with a time approximation. For the space approximation some standard finiteelement or finitedifference approxi mations are employed. We define a uniform grid with respect to time ωτ = ωτ ∪ {T } = {t n = nτ, n = 0,1,..., N , τN = T } and assume y n = y(x, t n ), t n = nτ, x ∈ Ω. For the approximate values, we use the same notation as for the exact ones. As the basic one, a purely implicit difference scheme is used. In this case, from (6) and (7) we move to the system n +1

ci

− ci n +1 n +1 n +1 = ∇(Di∇ ci + z i ui Fci ∇ϕ ), i = 1, 2,..., p, τ n

(12)

p

∇(ε∇ϕ

n +1

∑z c

n +1 i i ,

) = −F

x ∈ Ω, n = 0,1,..., N − 1.

(13)

i =1

Taking into account (9) and (10), we supplement (12) and (13) with the boundary conditions ϕ

n +1

=g

n +1

, x ∈ Γ1, ε

n +1

n +1

∂ϕ = 0, x ∈ Γ 2, ∂ν

(14)

n +1 ∂ ci n +1 ∂ϕ + z i ui Fci = 0, i = 1, 2,..., p, x ∈ ∂Ω. ∂ν ∂ν For problem (12)–(15), as in (11), we have

(15)

Di

∫

∫

M i ≡ cin+1dx = ci0dx, i = 1,2,..., p, n = 0,1,..., N − 1. Ω

(16)

Ω

For (16) to retain its conservatism, the appropriate space approximation must be chosen. The same applies to the property of monotonicity (adherence to the maximum principle); i.e., when purely implicit MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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VABISHCHEVICH, ILIEV γ2

1

γ3

x2

γ1 0 Fig. 1. Computational domain.

approximations are used for nonstationary diffusionconvection equations the monotonicity is not vio lated [9]. Scheme (12)–(15) has the first order time approximation. Proceeding from scheme (12) and (13), different time approximations can be used. Among the possible simplest generalizations a secondorder approximation can be mentioned. In this case, from (6) and (7) we pass to the system ⎛ cin+1 − cin c n+1 + cin c n+1 + cin ϕ n+1 + ϕ n ⎞ (17) = ∇ ⎜ Di∇ i + z i ui F i ∇ ⎟. τ 2 2 2 ⎝ ⎠ The main drawback of schemes (12) and (13) and (13) and (17) is associated with the problem nonlin earity at the new time layer, i.e., Eq. (12) and the boundary condition (15). The simplest modification with scheme (12) and (13) is based on the transition from (12) and (15) to n +1

n

n +1

n

− ci n +1 n +1 n (18) = ∇(Di∇ ci + z i ui Fci ∇ϕ ), i = 1, 2,..., p, τ n +1 n ∂c n +1 ∂ϕ (19) Di i + z i ui Fci = 0, i = 1, 2,..., p, x ∈ ∂Ω. ∂ν ∂ν For the linearized scheme (13), (14), (18), (19) the condition of conservatism of (16) holds true and the implementation involves the solution of individual elliptic problems. The second modification of the nonlinear scheme (12)–(15) is connected with the use of ci

− ci n +1 n n +1 (20) = ∇(Di∇ ci + z i ui Fci ∇ϕ ), i = 1, 2,..., p, τ n +1 n +1 ∂c n ∂ϕ (21) = 0, i = 1, 2,..., p, x ∈ ∂Ω. Di i + z i ui Fci ∂ν ∂ν In the case of the linearized scheme (13), (14), (20), and (21), the coefficients are taken from the pre vious time level and the system of equations itself remains coupled. By combining these two abovementioned linearizations we can construct a linearized scheme of the secondorder of accuracy when ci

⎛ ⎞ cin+1 − cin c n+1 + cin z i ui F n+1 n (ci ∇ϕ + cin∇ϕ n+1) ⎟ , = ∇ ⎜ Di∇ i + τ 2 2 ⎝ ⎠

(22)

n +1 ⎞ Di ∂(cin+1 + cin ) z i ui F ⎛ n+1 ∂ϕ n n ∂ϕ + + c c i i ⎜ ⎟ = 0, x ∈ ∂Ω. ∂ν ∂ν ∂ν ⎠ 2 2 ⎝ Scheme (13), (14), (22), and (23) is a linearized analog of scheme (13)–(15) and (17).

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Fig. 2. Triangulation.

4. TEST PROBLEM Various time approximations are tested on the solution of the model twodimensional problem. We restrict ourselves to the case of a binary electrolyte ( p = 2, z1 > 0, z 2 < 0) making the additional assump tion that z1u1 zu = − 2 2. D1 D2

In these conditions, under the appropriate nondimensionalization, the system of equations (6) and (7) takes the form ∂c+ (24) = ∇(∇c+ + c+∇ϕ), ∂t ∂c− (25) = ∇(∇ c− − c−∇ϕ), ∂t (26) εΔϕ = −c+ + c−, x ∈ Ω, where c+ = c1, c− = c2. The system of equations (24)−(26) is solved in the unit square Ω = {x = (x1, x2 ) 0 < xα < 1, α = 1,2}. Figure 1 shows parts of the boundary ∂Ω = γ 1 ∪ γ 2 ∪ γ 3 on which different boundary conditions are specified; at the same time, γ1 ∪ γ 2 = Γ1, γ 2 = Γ 2 and γ 2 = {x = (x1, x2 ) x ∈ ∂Ω, x2 = 1, 0 < x1 < 0.5}. In accordance with (8), we supplement the sim plest initial conditions (27) c+(x,0) = 1, c−(x,0) = 1, x ∈ Ω, which provide the electrical neutrality of the elec trolyte in the initial state. We take boundary conditions for the electric Fig. 3. Electric field potential ϕ : 0−1. potential (see (9)) in the form MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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VABISHCHEVICH, ILIEV

Fig. 4. Concentration c+ : 0.527−1.437.

As in (10), we assume

ϕ(x, t ) = 0, x ∈ γ 1, ϕ(x, t ) = 1, x ∈ γ 2, ∂ϕ ε (x, t ) = 0, x ∈ γ 3. ∂ν

(28) (29)

(∂∂νc + c ∂ϕ∂ν) (x,t) = 0, (∂∂νc − c ∂ϕ∂ν) (x,t) = 0, x ∈ ∂Ω. +

−

+

(30)

−

We use problem (24)–(30) for testing various time approximations. With respect to space, the same finiteelement approximation is used. Triangulation (the number of nodes is 1272, the number of triangles is 2420) is shown in Fig. 2 for the approximation of the electric potential ϕ and concentrations c+ and c– standard Lagrangian quadratic finite elements are used [10]. We present some data on the stationary solution of problem (24)–(30) at ε = 0.01 by the relaxation method (solution at a sufficiently large time T = 10 ). Figure 3 shows the isolines of the electric potential from 0 with step δϕ = (ϕ max − ϕ min ) 20 = 0.05. Similar data for the concentrations are presented in Figs. 4 and 5; the legends in the figures indicate the minimum and maximum values of the quantities. We trace the process dynamics using the electric field and concentrations at the section x1 = 0 at vari ous time instants with step δ t = 0.01. Figure 6 shows the potential values and Figs. 7 and 8 present the con centrations c+ and c–, respectively. Table 1. Accuracy of the purely implicit scheme ε 0.1

0.01

0.001

τ

ξ(ϕ)

ξ(c+ )

ξ(c− )

0.01 0.005 0.0025 0.01 0.005 0.0025 0.01 0.005 0.0025

0.00226053 0.00117047 0.00059161 0.01140351 0.00617958 0.00322601 0.03286959 0.01929922 0.01057592

0.00456508 0.00265707 0.00163707 0.00572091 0.00306698 0.00159160 0.00776209 0.00456180 0.00249399

0.00470744 0.00260960 0.00161245 0.00633652 0.00340213 0.00176302 0.00905152 0.00537773 0.00296142


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Fig. 5. Concentration c− : 0.672−1.826.

The fundamental impact on the solution properties is made by parameter ε. The reduction in ε brings about the flattening of the solution observed in the center of the computational domain and an increase in solution gradients near boundaries γ 1 and γ 2. 5. CALCULATION RESULTS The comparison of different schemes is carried out on the test problem (24)–(30). The accuracy is esti mated using the time approximations

∫

I n2(u) = (u n(x) − u(x, t n))2 dx, Ω

ξ(u) = maxI n(u), n = 1,2,..., N . n

ϕ 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

С+ 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x2 Fig. 6. Electric field potential ϕ(0, x2, t), t : 0−1, δt = 0.01.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x2 Fig. 7. Concentration c+ (0, x2, t), t : 0−1, δt = 0.01.


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VABISHCHEVICH, ILIEV C– 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x2 Fig. 8. Concentration c−(0, x2, t ), t : 0−1, δt = 0.01.

ξ

0.006

0.004

0.002

0

0.05

0.10

0.15

0.20 Time

Fig. 9. Error: the solid line is I n (ϕ), the dashed line is I n(c+ ), the dashedanddotted line is I n(c− )(ε = 0.01, τ = 0.005). n

Here, u (x) is the solution when some difference time approximations are employed, and u(x, t n) is the exact solution obtained by significantly more detailed time grids. With respect to space, the same finite

Table 2. Accuracy of the linearized scheme (1) ε 0.1 0.01 0.001

τ

ξ(ϕ)

ξ(c+ )

ξ(c− )

0.01 0.005 0.0025 0.01 0.005 0.0025 0.01 0.005 0.0025

0.00103774 0.00058519 0.00031500 0.03066031 0.01027917 0.00481860 931.763886 12713.0466 5591.45715

0.00515894 0.00300019 0.00176101 0.01038322 0.00498404 0.00249983 2238.02940 12156.7939 7622.22505

0.00517636 0.00297993 0.00174325 0.01125235 0.00520613 0.00259966 974.183650 11744.9361 10799.3506


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ξ 0.012 0.010 0.008 0.006 0.004 0.002

0

0.05

0.10

0.15

0.20 Time

Fig. 10. Error: the solid line is τ = 0.01, the dashed line is τ = 0.005, the dashedanddotted line is τ = 0.0025(ε = 0.01, I n(ϕ)).

ξ

0.010

0.005

0

0.05

0.10

0.15

0.20 Time

Fig. 11. Error of the linearized scheme (1): the solid line is I n (ϕ), the dashed line is I n(c+ ), the dashedanddotted line is I n (c − ) (ε = 0.01, τ = 0.005).

element approximation is used, which has been described above. In these conditions, I n(u) is an analog of the norm in L2(Ω) at time layer t n and ξ(u) is the discrete analog of C([0,T ], L2(Ω)). The solutions of the model problem (24)–(30) are compared for different time steps ( τ = 0.01, 0.005, 0.0025) while choosing T = 0.2 for ε = 0.1, 0.01, 0.001. Figure 9 shows the dependence of the error over time on using the purely implicit scheme (see (12)– (15)). The dependence of the calculation error of the electric potential on the time step can be traced in Fig. 10. A theoretical dependence is observed on τ (error O(τ) ). Data on the errors at different values of parameter ε on three computational grids are shown in Table 1. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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VABISHCHEVICH, ILIEV ξ 0.035 0.030 0.025 0.020 0.015 0.010 0.005

0

0.05

0.10

0.15

0.20 Time

Fig. 12. Error of the linearized scheme (1): the solid line is τ = 0.01, the dashed line is τ = 0.005, the dashedanddotted line is τ = 0.0025(ε = 0.01, I n(ϕ)).

ξ

0.006

0.004

0.002

0

0.05

0.10

0.15

0.20 Time

Fig. 13. Error of the linearized scheme (2): the solid line is I n (ϕ), the dashed line is I n(c+ ), the dashedanddotted line is I n (c − ) (ε = 0.01, τ = 0.005).

The second considered time approximation is based on the linearization of type (18), (19). Its imple mentation is related to the solution of individual linear problems for the potential and concentrations. To begin the calculations, it is necessary to specify the initial distribution of the electric potential. In accor dance with (26), we assume

εΔϕ0 = −c+0 + c−0, x ∈ Ω.

(31)

Figure 11 shows the dependence of the error over time when the linearized scheme is used and Fig. 12 shows its dependence on the time step. A comparison with the results of the calculations by a nonlinear MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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ξ 0.012 0.010 0.008 0.006 0.004 0.002

0

0.05

0.10

0.15

0.20 Time

Fig. 14. Error of the linearized scheme (2): the solid line is τ = 0.01, the dashed line is τ = 0.005 the dashedanddotted line is τ = 0.0025(ε = 0.01, I n(ϕ)).

purely implicit scheme (see Figs. 9 and 10) demonstrates a significant drop in the accuracy of the approx imate solution. Moreover, for sufficiently small values of the coefficient ε the linearized scheme diverges. In particular, as shown by the data in Table 2, this linearized scheme does not converge at ε = 0.001. A more promising linearization for the numerical solution of the boundaryvalue problems for the sys tem of Nernst–Planck equations is based on the associated solution of equations for concentration and electric potential. In this case (see (20) and (21)), the concentration is taken from the lower time layer and the potential is assigned to the upper one. The error of this difference scheme is illustrated by Fig. 13 (the errors for the concentrations are close to each other). The reduction in the error with a decrease in param eter ε can be traced in Fig. 14. The accuracy of the results are summarized in Table 3. Comparison with the results obtained by the purely implicit scheme (Figs. 9 and 10 and Table 1) shows that the considered linearized scheme yields results in a similar accuracy. Here, the following explanations can be given. Table 3. Accuracy of the linearized scheme (2) ε

0.1

0.01

0.001

τ

ξ(ϕ)

ξ(c+ )

ξ(c− )

0.01

0.00214027

0.00451117

0.00560687

0.005

0.00110700

0.00242965

0.00297361

0.0025

0.00056299

0.00151363

0.00178570

0.01

0.01115596

0.00555310

0.00774706

0.005

0.00604812

0.00297024

0.00411108

0.0025

0.00315407

0.00154017

0.00211744

0.01

0.03261027

0.00751146

0.01152227

0.005

0.01894858

0.00441216

0.00669605

0.0025

0.01039285

0.00241903

0.00362478


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VABISHCHEVICH, ILIEV ξ 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005

0

0.05

0.10

0.15

0.20 Time

Fig. 15. Error of the symmetric scheme: the solid line is I n (ϕ), the dashed line is I n(c+ ), the dashedanddotted line is I n (c − ) (ε = 0.01, τ = 0.005).

The linear prototype of Eqs. (24) and (25) is the nonstationary convectiondiffusion equation. The lin earized scheme (20) and (21) can be interpreted as an explicitimplicit scheme for the convectiondiffu sion equation [9] when the convective terms are taken from the lower time layer and diffusion is taken from the upper one. Explicitimplicit schemes for the nonstationary convectiondiffusion equation are classi fied as unconditionally stable; moreover, stability is proved in a stronger norm than for the standard purely implicit schemes [9, 11]. In view of this, it is not surprising that the linearized scheme (20) and (21) yields good results. The results are presented for the solution of the model problem (24)–(30) using the schemes of the sec ondorder time approximation. In the computations, the initial state of the electric field is calculated from the solution of the respective boundaryvalue problem for Eq. (31). The time dependence of the error when the nonlinear scheme (17) is employed is shown in Fig. 15. The reduction in the calculation error of the electric potential with the decrease in the time step is illustrated Table 4. Accuracy of the symmetric scheme ε

0.1

0.01

0.001

τ

ξ(ϕ)

ξ(c+ )

ξ(c− )

0.01

0.00059758

0.00516594

0.00507193

0.005

0.00023621

0.00315764

0.00312166

0.0025

0.00009015

0.00192925

0.00191551

0.01

0.00361763

0.00449925

0.00406404

0.005

0.00207446

0.00287323

0.00273384

0.0025

0.00089409

0.00183267

0.00179109

0.01

0.01387564

0.00518872

0.00402675

0.005

0.00618539

0.00244920

0.00185191

0.0025

0.00363811

0.00141304

0.00116302


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ξ 0.0040 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0

0.05

0.10

0.15

0.20 Time

Fig. 16. Error of the symmetric scheme: the solid line is τ = 0.01, the dashed line is τ = 0.005 the dashedanddotted line is τ = 0.0025 (ε = 0.01, I n (ϕ)).

ξ 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005

0

0.05

0.10

0.15

0.20 Time

Fig. 17. Error of the symmetric scheme: the solid line is I n (ϕ), the dashed line is I n(c+ ), the dashedanddotted line is I n (c − ) (ε = 0.01, τ = 0.005).

by Fig. 16. Integrated data on errors at different values of parameter ε on three time grids are presented in Table 4. In addition to a significant improvement in the accuracy in the comparison with a purely implicit scheme (see Figs. 9 and 10 and Table 1), we also observe nonmonotonicity of the error. The theoretical dependence on τ (error O(τ ) ) is violated on the initial segment and is due to the inconsistency of the ini tial and boundary conditions. 2


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VABISHCHEVICH, ILIEV ξ 0.0040 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0

0.05

0.10

0.15

0.20 Time

Fig. 18. Error of the symmetric scheme: the solid line is τ = 0.01, the dashed line is τ = 0.005, the dashedanddotted line is τ = 0.0025 (ε = 0.01, I n (ϕ)).

The linearized scheme of the secondorder accuracy (22) (see Figs. 17 and 18 and Table 5) yields an approximate solution with about the same accuracy as the symmetric scheme (17) (compare with the cor responding graphical and tabular data in Figs. 15 and 16 and in Table 4). It is important that in contrast to (17), scheme (22) is linear and the employed linearization makes sufficient allowance for the quadratic nonlinearity of the problem. 6. CONCLUSIONS The results of the numerical investigation into the model twodimensional problem for a binary elec trolyte based on the initial boundaryvalue problem for the system of the Nernst–Planck equations with a quadratic nonlinearity can be summarized as follows. 1. The purely implicit scheme for the considered nonlinear problem is stable and has the first order time approximation. Its main drawback consists in its computational implementation, as at the new time level, it is necessary to solve a coupled system of nonlinear grid elliptic equations. 2. The stability and accuracy characteristics close to the purely implicit scheme are exhibited by the linearized scheme for the Nernst–Planck equations when, in the equation for the concentration of elec Table 5. Accuracy of the linearized symmetric scheme ε 0.1

0.01

0.001

τ

ξ(ϕ)

ξ(c+ )

ξ(c− )

0.01 0.005 0.0025 0.01 0.005 0.0025 0.01 0.005 0.0025

0.00059748 0.00023620 0.00009015 0.00361158 0.00207347 0.00089396 0.01407924 0.00627989 0.00366601

0.00519185 0.00316467 0.00193112 0.00475325 0.00294275 0.00185128 0.00687275 0.00306603 0.00159501

0.00504397 0.00311408 0.00191351 0.00387423 0.00266783 0.00177199 0.00456439 0.00190144 0.00107175


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trically charged particles, an explicit approximation with respect to concentration and an implicit approx imation with respect to the electric field are employed for the nonlinear terms of the equations which describe the electric drift of the particles. The second possibility, involving the implicit approximations with respect to concentration and explicit approximations with respect to the electric field leads, to lin earized schemes that are much worse in terms of accuracy and stability. 3. A linearized scheme of the secondorder time approximation is constructed, which takes into account the quadratic nonlinearity of the considered problems for the Nernst–Planck equations. This scheme in terms of accuracy and stability is almost as good as the standard symmetric nonlinear Crank– Nicolson difference scheme but offers decisive advantages in computational implementation. REFERENCES 1. J. S. Newman and K. E. ThomasAlyea, Electrochemical Systems (Wiley, New York, 2004). 2. A. J. Bard, M. Stratmann, E. J. Calvo, et al., Encyclopedia of Electrochemistry, Interfacial Kinetics and Mass Transport, Vol. 2 (Wiley, New York, 2003). 3. C. F. Wallgren, F. H. Bark, R. Eriksson, D. S. Simonsso, J. Perssont, R. I. Karlsson, “Mass Transport in a Weakly Stratified Electrochemical Cell,” J. Appl. Electrochem. 26 (12), 1235 (1996). 4. V. Fila and K. Bouzek, “A Mathematical Model of Multiple Ion Transport across an IonSelective Membrane under Current Load Conditions,” J. Appl. Electrochem. 33 (8), 675 (2003). 5. M. Buoni and L. Petzold, “An Efficient, Scalable Numerical Algorithm for the Simulation of Electrochemical Systems on Irregular Domains,” J. Comp. Phys. 225 (2), 2320 (2007). 6. A. Prohl and M. Schmuck, “Convergent Discretizations for the Nernst–Planck–Poisson System,” Num. Mat. 111 (4), 591 (2009). 7. A. A. Samarskii, Theory of Difference Schemes (Nauka, Moscow, 1989) 8. A. Friedman, Partial Differential Equations of Parabolic Type (PrenticeHall, Englewood Cliffs, 1964). 9. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for the Solution of ConvectionDiffusion Problems (Editorial URSS, Moscow, 1999) [in Russian]. 10. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, (NorthHolland, Amsterdam, 1978). 11. A. A. Samarskii and P. N. Vabishchevich, ExplicitImlicit Difference Schemes for ConvectionDiffusion Problems. Recent Advances in Numerical Methods and Applications II, Ed. by O.P. Iliev, M.S. Kashchiev, S.D. Margenov, B.H. Sendov and P.S. Vassilevski (World Scientific, Singapore, 1999).

Translated by I. Pertsovskaya


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2013