An exponentially-fitted method
for singularly perturbed,
one-dimensional, parabolic problems
J.I. Ramos
Room I-320-D, E.T.S. Ingenieros Industriales, Universidad de Malaga, Malaga 29013, Spain
Abstract
An exponentially-fitted method for singularly perturbed, one-dimensional, linear, convection–diffusion–reaction equations in equally-spaced grids is presented. The method is based on the implicit discretization of the time derivative, freezing of the coefficients of the resulting ordinary differential equations at each time step, and the analytical solution of the resulting convection–diffusion differential operator. This solution is of exponential type and exact for steady, constant-coefficients convection– diffusion equations with constant sources. By means of three examples, it is shown that this method provides uniformly convergent solutions with respect to both the time step and the small perturbation parameter, and these solutions are more accurate and exhibit a higher order of convergence than those obtained with an upwind finite difference scheme in a piecewise uniform mesh that is boundary-layer resolving.
Ó 2004 Elsevier Inc. All rights reserved.
Keywords:Exponentially-fitted method; Singularly perturbations; Parabolic equations; Advection– reaction–diffusion equations
1. Introduction
It is well known that the fluid dynamics and heat/mass transfer equations may exhibit sharp boundary layers at high Reynolds and Peclet numbers, respectively, so that the use of standard central finite difference methods or
E-mail address:[email protected](J.I. Ramos).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.046
finite element techniques with piecewise polynomial basis functions often yields numerical solutions with non-physical spurious oscillations near these layers [1–5]. An analogous comment applies to the numerical solution of the semi-conductor device equations [6]. In order to avoid these non-physical oscilla-tions, most investigators have resorted to methods based on either upwinding of the convection/advection terms or exponentially-fitted techniques. Expo-nentially-fitted methods are based on the idea of approximating a flux density locally by a either a constant or a linear function of the independent spatial variable, whereas upwinding techniques are based on local polynomial approximations. Since flux densities are better behaved than the local approximations used with upwinding techniques, exponentially-fitted methods have been very popular in the numerical simulation of semiconductor devices, while upwind techniques have been frequently used in computational fluid dynamics and heat transfer.
Most of the exponentially-fitted methods are extensions of the techniques developed by Allen and Southwell [7], IlÕin [8] and Scharfetter and Gummel [9] who argued that, since the solution of a linear homogeneous ordinary differ-ential equation with constant coefficients is expondiffer-ential, the difference scheme should be chosen to be exact for that exponential. IlÕin [8] showed that his exponentially-fitted scheme has an error bound of the form Ch where C is independent of bothhand the inverse of the Reynolds number, wherehis the grid spacing, for one-dimensional, steady, advection–diffusion equations without source terms.
Using ideas from generalized compact-operator implicit methods, El-Mist-ikawy and Werle [10] developed an exponentially-fitted scheme for steady convection-diffusion equations with an error bound of the formCh2whereCis
independent of bothh and the inverse of the Reynolds number. The schemes proposed by Allen and Southwell [7], IlÕin [8], Scharfetter and Gummel [9] and El-Mistikawy and Werle [10] are examples of a broader class of locally-exact methods which can be derived by means of the (local) Green function on two adjacent non-overlapping intervals, upon making certain assumptions on the coefficients of the ordinary differential equation. Alternatively, one may derive the schemes of Allen and Southwell [7], Scharfetter and Gummel [9] and IlÕin [8] by considering three-point intervals, freezing the coefficients of the differ-ential equation at the midpoint, solving the resulting steady convection–dif-fusion equation analytically and imposing that this solution yields the nodal values at the two end points and at the midpoint of each interval [11]. This procedure results in a tridiagonal matrix for the nodal values and is exact for linear, constant-coefficients ordinary differential equations with constant right-hand sides.
Recently, much effort has been placed on the development of numerical methods for the solution of singular perturbation problems that are uniformly convergent with respect to the perturbation parameter, e.g., the inverse of the
Reynolds number in high speed flows [4,5,12], i.e., numerical methods that are robust in the sense that the error in the approximation does not deteriorate as the singular perturbation parameter is decreased towards zero. The key idea of these methods is the use of either piecewise uniform meshes which are appropriately condensed in the boundary layer region, e.g., ShishkinÕs meshes [13], or exponentially-graded meshes [14]. Alternatively, one may use thep=hp
version of the finite element method in appropriately designed meshes so that, in principle, one could achieve exponential rates of convergence for smooth domains [15].
In this paper, we present an exponentially-fitted method for singularly perturbed, one-dimensional (advection–reaction–diffusion) parabolic prob-lems, and show its uniform convergence in the perturbation parameter. The method is based on the discretization of the time derivative at each time step, the freezing of the spatially dependent coefficients, and the piecewise analytical solution of the resulting (advection – diffusion–reaction) ordinary differential in an equally-spaced mesh. Such a solution results in a three-point, exponentially-fitted finite difference equation which can easily be solved. The approach presented here is to be contrasted with that proposed by the author [11] who discretized the time derivative and included the resulting terms in the ordinary differential equation. It should also be contrasted with that proposed by Cla-vero et al. [16] who also discretized the time derivatives but employed upwind differences in a piecewise uniform ShishkinÕs mesh and obtained a uniformly convergent method with respect to the perturbation parameter, i.e., the diffu-sion coefficient, and the time step.
The paper has been organized as follows. In Section 2, we present the exponentially-fitted method for singularly perturbed, one-dimensional para-bolic problems, and show that this method reduces to that of IlÕin [8] for steady-state, convection–diffusion equations in the absence of source terms. In Section 3, the exponentially-fitted method is applied to the three, singularly-perturbed, one-dimensional parabolic problems considered by Clavero et al. [16] and comparisons are made with other solutions based on upwind differ-ences in piecewise uniform meshes. A summary of the main conclusions puts an end to the paper.
2. Exponentially-fitted method
Consider the following singularly perturbed, one-dimensional (linear) par-abolic problem of the advection–diffusion–reaction type
ou
otþaðx;tÞ
ou
oxþbðx;tÞu¼
o2u
ox2þfðx;tÞ; 0<x<1; t>0; ð1Þ
uð0;tÞ ¼uð1;tÞ ¼0; tP0; ð2Þ
uðx;0Þ ¼u0ðxÞ; 06x61; ð3Þ
where 0< 1 is the diffusion coefficient or perturbation parameter,xandt
denote the spatial coordinate and time, respectively,u is the dependent vari-able,aðx;tÞis the speed, andfðx;tÞ bðx;tÞuis the reaction term.
Discretization of the time derivative by means of the backward Euler method in Eq. (1) yields the following linear ordinary differential equation at each time step
aðx;tÞdu
dx
d2u
dx2¼fðx;tÞ bðx;tÞu
uun
Dt ; 0<x<1; t>0; ð4Þ
whereun¼uðx;tnÞ, Dtis the time step, the superscript n denotes thenth time
level, i.e.,tn¼nDt,ttnþ1 anduunþ1.
The linear ordinary differential Eq. (4) cannot, in general, be solved ana-lytically because of the dependence ofa,b,f andun on the spatial coordinate.
However, if the spatial domain [0,1] is divided intoN equally spaced intervals of size equal toh, i.e.,Nh¼1, and one considers the intervalðxi1;xiþ1Þand the
coefficients of Eq. (4) are evaluated at the midpoint of each interval, then one obtains the following linear ordinary differential equation
ai
du
dx
d2u
dx2¼Sifibiui
uiuni
Dt ; xi1<x<xiþ1; t>0; ð5Þ
whose analytical solution can be expressed as
u¼AiþBiexpðkiðxxiÞÞ þaiðxxiÞ; if ai¼0; xi1<x<xiþ1; t>0; ð6Þ
u¼ Si
2ðxxiÞ
2
þCiðxxiÞ þDi; if ai¼0; xi1<x<xiþ1; t>0; ð7Þ
whereki¼ai,ai¼aSii, and the constantsAi,Bi,CiandDican be determined from
the conditions
uðxi1Þ ¼ui1; uðxiÞ ¼ui; uðxiþ1Þ ¼uiþ1: ð8Þ
These conditions result in the following tridiagonal system of algebraic equations
h2ðui12uiþuiþ1Þ
tanhh h
ai
2hðuiþ1ui1Þ
tanhh h
1
Dtð1þbiDtÞui
¼ tanhh
h
1
Dtðu n
i þfiDtÞ; i¼2;3;. . .;N; ð9Þ
h2ðui12uiþuiþ1Þ
1
Dtð1þbiDtÞui¼
1
Dtðu n
i þfiDtÞ; i¼2;3;. . .;N;
ð10Þ
if ai¼0, where h¼aih=2 is half the mesh Reynolds number and u1¼uNþ1¼0.
The derivation of the exponentially-fitted method presented above differs substantially from that of the author [11] who solved the following linear or-dinary differential equation
bi
þ 1
Dt
uþai
du
dx
d2u
dx2 ¼Tifiþ
un i
Dt; xi1<x<xiþ1; t>0;
ð11Þ
which can be obtained from Eq. (4) and incorporates the effects of the reaction and the time derivative through the first two terms.
Whenou
ot¼0,ais constant,b¼0, andf ¼0, it can be easily shown that the
method presented above (cf. Eq. (9)) is identical to IlÕinÕs uniformly convergent scheme for steady, one-dimensional, advection–diffusion equations without sources terms. IlÕinÕs method has been derived by Roos [17] by several tech-niques including freezing the coefficients of the differential equation, compact exponentially-fitted methods, exact difference techniques, collocation, finite volumes, polynomial-conforming Petrov–Galerkin finite elements, exponential Petrov–Galerkin finite elements, explicit Galerkin techniques and mixed finite elements. In addition, Eq. (9) indicates that forh 1, the diffusion terms are negligible compared with the convective ones and the advection–diffusion operator tends to a convective one. On the other hand, forjhj 1, convection and diffusion are of the same order of magnitude.
It must be noted that the derivation presented above was based on con-sidering the advection and diffusion operators and freezing the time derivative and source or reaction terms. If instead of Eq. (5), one considers Eq. (11), one will also obtain exponentially-fitted methods that account for the advection, diffusion, reaction and transient terms [11]. These methods, however, have been found not to perform as well as the fitted method presented above whenis very small because of the roots of the characteristic polynomial of the resulting linear ordinary differential equation were found to be nearly insensitive to the time step.
3. Results
In order to assess both the accuracy and the order of the exponentially-fitted method presented in this paper for singularly perturbed, one-dimensional parabolic problems, we have applied it to the following three examples.
Example 1.This example corresponds toaðx;tÞ ¼1,bðx;tÞ ¼0, and the exact solution given by
ueðx;tÞ ¼expðtÞ C1
þC2xexp
1x
; ð12Þ
whereC1¼expð1Þ, C2¼1C1, and the values of u0ðxÞand fðx;tÞ can be
determined from Eqs. (3) and (1), respectively.
Since this problem has an analytical solution, the nodal errors can be cal-culated as
eN;Dt
ðxi;tnÞ ¼ jueðxi;tnÞ uNðxi;tnÞj; ð13Þ
and the maximum nodal errors are
EN;Dt
¼maxi;n e
N;Dt
; ð14Þ
for 06t61.
It must be noted that in all the examples considered in this paper, we em-ployedDt¼0:1 withN¼16, and that whenNis multiplied by 2,Dtis divided by two.
The -uniform maximum nodal error is defined as EN;Dt¼max
EN;Dt, and
the numerical-uniform rate of convergence is given by
pN ¼
logðEN;Dt
=E2N;Dt=2Þ
log 2 ; ð15Þ
which is presented in Table 1.
Example 2.This example corresponds toaðx;tÞ ¼2x2,bðx;tÞ ¼x,u
0ðxÞ ¼0
andfðx;tÞ ¼10t2expðtÞxð1xÞand does not have an exact solution.
In this example as well as in the next one, the pointwise errors were esti-mated as
eN;Dtðxi;tnÞ ¼ juNðxi;tnÞ u2Nðxi;tnÞj; ð16Þ
whereas the maximum nodal errors and the numerical-uniform rate of con-vergence were determined as in Example 1 for 06t63, and the results are
presented in Table 2.
Example 3. This example corresponds to aðx;tÞ ¼2x2, bðx;tÞ ¼x2þ1þ
cosðpxÞ,u0ðxÞ ¼0 andfðx;tÞ ¼sinðpxÞand does not have exact solution. The
pointwise and maximum nodal errors and the numerical -uniform rate of convergence were determined as in Example 2 for 06t61, and the results are
The results presented in Tables 1–3 clearly indicate that the exponentially-fitted method presented in this paper is -uniformly convergent. For a fixed value of, the pointwise numerical errors and the maximum nodal errors de-crease whereas the convergence order and the -uniform rate of convergence increase, in general, as the number of grid points increases.
For Examples 1 and 2, the results presented in this paper indicate that the maximal nodal errors and the-uniform rate of convergence of the exponen-tially-fitted method presented in this paper are smaller and larger, respectively, than those obtained by means of upwind differences in piecewise uniform meshes [16]. Table 2 shows that, in some cases, the exponentially-fitted method has a convergence order greater than one. Similar comments apply to the
Table 1
Maximum nodal errors and order of the fitted method for Example 1
N¼16 N¼32 N¼64 N¼128 N¼256 N¼512 N¼1024
20 2.8401e)4 1.4735e)4 7.5040e)5 3.7857e)5 1.9011e)5 9.5245e)6 4.7660e)6
0.9467 0.9735 0.9871 0.9938 0.9971 0.9989
22 3.7e)3 1.9e)3 9.6559e)4 4.8719e)4 2.4472e)4 1.2264e)4 6.1393e)5
0.9623 0.9748 0.9869 0.9933 0.9966 0.9983
24 9.4e)3 4.9e)3 2.5e)3 1.2e)3 6.2776e)4 3.1471e)4 1.5757e)4
0.9577 0.9728 0.9855 0.9925 0.9962 0.9981
26 1.25e)2 6.5e)3 3.3e)3 1.7e)3 8.3650e)4 4.1953e)4 2.1009e)4
0.9567 0.9722 0.9841 0.9915 0.9956 0.9978
28 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5477e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9907 0.9952 0.9976
210 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
212 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
214 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
216 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
218 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
220 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
222 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
224 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
226 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
EN;Dt 1.28e)2 6.6e)3 3.4e)3 1.7e)3 8.5473e)4 4.2881e)4 2.1477e)4
pN 0.9585 0.9726 0.9832 0.9906 0.9951 0.9976
EN;Dt[16] 7.2411e)2 4.9395e)2 3.0647e)2 1.7952e)2 1.0184e)2 5.6568e)3
results presented in Table 3 except forN ¼128 and 256 which show that even though the exponentially-fitted method yields smaller maximum nodal errors than upwind differences in piecewise uniform meshes [16], its accuracy is slightly lower than that of the latter.
For the problems considered in this paper, there is a boundary layer atx¼1 the thickness of which is on the order of. This boundary layer is, therefore, not resolved by the fitted method presented in this paper for <N1; however,
the method still provides accurate solutions. By way of contrast, in a piecewise uniform Shishkin mesh, the interval [0,1] is divided into two non-overlapping subintervals½0;1rand½1r;1withðNþ1Þ=2 grid points in each, where
r¼minf1
2;mlogNg,aðxÞ>a>0 andmP 1
a, each subinterval is divided into
Table 2
Maximum nodal errors and order of the fitted method for Example 2
N¼16 N¼32 N¼64 N¼128 N¼256 N¼512
20 4.7605e)4 2.4599e)4 1.2551e)4 6.3441e)5 3.1898e)5 1.5994e)5
0.9525 0.9708 0.9843 0.9919 0.9971
22 1.8e)3 9.5501e)4 4.9354e)4 2.5131e)4 1.2687e)4 6.3744e)5
0.9227 0.9524 0.9737 0.9862 0.9929
24 4.2e)3 1.6e)3 8.7968e)4 4.5994e)4 2.3538e)4 1.1908e)4
1.3915 0.8779 0.9354 0.9666 0.9830
26 9.1e)3 3.7e)3 1.1e)3 6.0029e)4 3.1797e)4 1.6362e)4
1.2974 1.7234 0.9023 0.9168 0.9585
28 1.01e)2 4.9e)3 2.4e)3 9.5067e)4 3.0865e)4 1.6717e)4
1.0387 1.0273 1.3455 1.6230 0.8847
210 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.1385e)4 2.3940e)4
1.0325 0.9747 0.9852 1.0503 1.3584
212 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2044e)4
1.0325 0.9747 0.9840 0.9921 0.9973
214 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
216 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
218 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
220 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
222 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
224 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
226 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
1.0325 0.9747 0.9840 0.9921 0.9961
EN;Dt 1.01e)2 4.9e)3 2.5e)3 1.3e)3 6.3968e)4 3.2071e)4
pN 1.0325 0.9747 0.9840 0.9921 0.9961
EN;Dt[16] 3.6999e)2 2.0815e)2 1.127e)2 5.8722e)3 3.0834e)3 1.6235e)3
equally-spaced meshes and upwind differences are used to discretize the advection terms in Eq. (4). For a¼1, m¼1, ¼106 and N¼1024, r is
about 3 and, therefore, the upwind difference method in piecewise uniform meshes developed by Clavero et al. [16] has about one sixth of the total number of grid points in the boundary layer and is a boundary layer-resolving method, albeit with a first-order accurate upwind scheme for convection. The expo-nentially-fitted method presented in this paper is not a boundary-layer resolving one for¼106 andN ¼1024, but, for Examples 1 and 2, is more
accurate than upwind difference methods in piecewise uniform meshes. Therefore, it may be conjectured that the numerical simulation of singularly perturbed, one-dimensional parabolic problems requires that either the
Table 3
Maximum nodal errors and order of the fitted method for Example 3
N¼16 N¼32 N¼64 N¼128 N¼256 N¼512
20 2.2e)3 3.4e)3 2.0e)3 1.1e)3 5.6229e)4 2.8725e)4
0.6399 0.7906 0.8853 0.9397 0.9690
22 8.1e)3 4.8e)3 2.7e)3 1.4e)3 7.2409e)4 3.6748e)4
0.7456 0.8540 0.9203 0.9581 0.9785
24 1.31e)2 7.6e)3 4.3e)3 2.3e)3 1.2e)3 6.0731e)4
0.7845 0.8398 0.8977 0.9424 0.9691
26 2.20e)2 1.26e)2 6.8e)3 3.6e)3 1.9e)3 9.5754e)4
0.8091 0.8963 0.9075 0.9428 0.9677
28 2.43e)2 1.60e)2 9.4e)3 5e)3 2.5e)3 1.3e)3
0.6041 0.7574 0.9230 0.9885 0.9934
210 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.2e)3 1.6e)3
0.5969 0.6867 0.7573 0.8604 1.0040
212 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8017 0.8580
214 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
216 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
218 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
220 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
222 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
224 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
226 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
0.5969 0.6866 0.7519 0.8016 0.8535
EN;Dt 2.43e)2 1.60e)2 1.0e)2 5.9e)3 3.4e)3 1.9e)3
pN 0.5969 0.6866 0.7519 0.8016 0.8535
EN;Dt[16] 3.5868e)2 2.5732e)2 1.6561e)2 1.0029e)2 5.6621e)3 2.0186e)3
boundary layer be resolved even with upwind schemes for convection or the technique accounts for the exponential behavior near boundary layers.
4. Conclusions
An exponentially-fitted method for singularly perturbed, one-dimensional parabolic problems of the advection–diffusion–reaction type based on the (first-order) implicit discretization of the time derivatives, freezing the coeffi-cients of the resulting ordinary differential equations, and analytical solutions of the resulting convection–diffusion operator has been developed for equally-spaced grids. It has been found that this method provides uniformly conver-gent solutions with respect to both the small perturbation parameter and the time step. It has also been found that the exponentially-fitted method is, for the three linear problems considered in this paper, more accurate and has a higher order of convergence than upwind finite differences schemes in piecewise uni-form meshes which are boundary-layer resolving, despite the fact that the fitted method is not a boundary-layer resolving one for small values of the pertur-bation parameter and/or when a small number of grid points is used in the calculations.
It has been shown that the fitted method presented in this paper reduces to the Allen–Southwell–IlÕin scheme for steady convection–diffusion equations without source terms. The method is not exact for constant-coefficient linear ordinary differential equations of the advection–reaction–diffusion type if the reaction terms are linear functions of the dependent variable, because the method freezes the reaction terms and only considers the analytical solution to the convection–diffusion operator.
Acknowledgements
This research was partially financed by Project BFM2001-1902 from the Ministerio de Ciencia y Tecnologıa of Spain and fondos FEDER.
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