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- *Corresponding Author:
- Carlos Henrique Marchi

Federal University of Paraná (UFPR)

Department of Mechanical Engineering Curitiba, PR, Brazil

**Tel:**554133613126

**Fax:**554133613701

**E-mail:**[email protected]

**Received** March 20, 2013; **Accepted** May 22, 2013; **Published** May 30, 2013

**Citation:** Marchi CH, Germer EM (2013) Effect of Ten CFD Numerical Schemes on Repeated Richardson Extrapolation (RRE). J Appl Computat Math 2:128. doi: 10.4172/2168-9679.1000128

**Copyright:** © 2013 Marchi CH, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

The main objective of this work is to evaluate the performance of RRE in reducing the discretization error when associated with ten types of CFD numerical schemes of first, second and third orders of accuracy. The onedimensional advection-diffusion equation is solved with the finite volume method, for five values of the Peclet number (Pe), with uniform grids of 5 to 23,914,845 volumes, allowing for up to 14 RRE. Results are obtained for temperature at the center of the domain, average of the temperature field, and heat transfer rate. It was found that: (1) RRE is extremely effective in reducing the discretization error for all the variables, numerical schemes and Pe, reaching an order of accuracy of up to 18.9; and (2) The second-order central difference scheme together with RRE is the one that presents the smallest error for the dependent variable.

Richardson extrapolation; Advection-diffusion equation; High-order scheme; Discretization error; Order of accuracy; Verification

The Repeated Richardson Extrapolation (RRE) was created [1,2] for the purpose of reducing and estimating the discretization error [3] of numerical solutions. To use RRE requires having the numerical solution of the variable of interest in three or more grids with different numbers of nodes, obtained by the finite volume or finite difference methods, for example. The importance of RRE in the area of CFD (Computational Fluid Dynamics) can be understood upon analyzing the optimal results of works [4-7]. For example, the minimum value of the stream function obtained in [5] was -0.11894 with two Richardson extrapolations based on 100x100, 120x120 and 140x140 grids. This result refers to the solution of the 2D Navier-Stokes equations of the lid-driven square cavity problem for the Reynolds number 1,000. This result has a smaller error than that obtained on a 1024x1024 grid without extrapolation. The above cited works solved two-dimensional problems using only the central difference scheme of second-order accuracy with up to 9 RRE. The following questions remain unanswered in the current literature:

a) It is known that without RRE, the higher the order of accuracy of the scheme the smaller the discretization error of the dependent variable for the same sufficiently fine grid. Could this also be the case when the schemes are associated with RRE?

b) Does any scheme, even of first-order accuracy, significantly reduce the discretization error when associated with RRE?

c) Can two schemes with the same order of accuracy result in very different discretization errors when associated with RRE?

d) Does the solution of advection-diffusion problems together with RRE result in reductions of the discretization error as large as in purely diffusive problems such as those obtained in [8,9]?

e) What is the effect of the various schemes together with RRE on the reduction of the discretization error of secondary variables, which are obtained by additional numerical approximations on the solution of the dependent (primary) variable?

f) What is the Peclet number effect on the reduction of the error of a scheme together with RRE?

To answer the above questions this study tests, along with RRE, ten types of schemes of first, second and third orders of accuracy [10], almost all of which are widely used in CFD. Grids with up to millions of nodes are used, resulting in up to fourteen RRE. For five values of the Peclet number, results are obtained for the temperature at the center of the calculation domain (dependent variable) and two secondary variables: average of the temperature field and the heat transfer rate. Finding the answers to these questions is irrefutable evidence of the importance of this work, because knowing what the best combination of a numerical scheme with RRE is enables one to obtain numerical solutions to practical problems of CFD with smaller error and at lower computational cost (RAM and CPU time).

Obviously, to perform this work, the ideal situation would be to solve three-dimensional or at least two-dimensional problems. However, as we will see in the results section, this could not be done with the computers available today, i.e., use millions of nodes in each direction; such a grid size would be required to adequately characterize the error and its orders of variables with the various schemes, associated or not with the RRE. Therefore, in this work, a one-dimensional advectiondiffusion problem is solved using the finite volume method on uniform grids. Nevertheless, RRE can be used in two-dimensional problems when using coarser grids and fewer REs than in the present work, as was done in [4-9]. Although RRE can also be used in three-dimensional problems, we are unaware of studies that have done this. Preliminary results of this work have been published in [11]. A complete and detailed description of this work can be found in [12].

The mathematical model considered in this work consists of the one-dimensional steady state advection-diffusion equation with Dirichlet boundary conditions. This equation is widely used in testing new numerical models [13] and is defined by

(1)

where *Pe* is the Peclet number, x is the coordinate direction, and T is the temperature. The analytical solution of Eq. (1) is T(x)=(exp(xPe)-1)/ (exp(*Pe*)-1). The variables of interest are: (1) the temperature at the center of the domain, i.e., at x=½, represented by *T _{c}*; (2) the average temperature field, represented by

The variables *Tm* and q are defined mathematically by

, (2)

where S=1 m, k=1 W/m.K and a=1 m^{2} represent, respectively, the length of the calculation domain, the thermal conductivity, and the area of heat exchange.

**Numerical solutions without extrapolation**

With the finite volume method [10], integrating Eq. (1) on the generic control volume P of **Figure 1**, one obtains

(3)

To approximate *T* on the west (*w*) and east (*e*) faces of each P volume, which is required in Eq. (3), the following advection schemes were used, indicated together with their orders of accuracy:

• UDS-1: Upwind Differencing Scheme (first order) [10]

• Alpha (first order) [14]

• CDS-2: Central Differencing Scheme (second order) [10]

• UDS-2: Upwind Differencing Scheme (second order) [10]

• QUICK: Quadratic Upstream Interpolation for Convective Kinematics (third order) [10,12]

• ADS: Adaptable Difference Scheme (second order) [15]

• TVD: Total Variation Diminishing (Superbee type, second order) [15].

To approximate the first-order derivative of *T* on the west (*w*) and east (*e*) faces of each P volume, which is required in Eq. (3), the following diffusion schemes were used, indicated together with their orders of accuracy:

• CDS-2: Central Differencing Scheme (second order) [10]

• CDS-4: Central Differencing Scheme (fourth order). In this case, four types of third-order schemes were also used to evaluate the firstorder derivatives of Eq. (3) on the faces of the two boundaries and on the face closest to each boundary, to which the CDS-4 scheme cannot be applied [12]. To approximate T and its first-order derivative on the west (*w*) and east (*e*) faces of each P volume, which is required in Eq. (3), the following advection-diffusion schemes were used, indicated together with their orders of accuracy:

• WUDS: Upstream-Weighted Differencing Scheme (second order) [15]

• PLDS: Power-Law Differencing Scheme (second order) [10].

Using the seven advection schemes and two diffusion schemes mentioned above, eight advection-diffusion schemes were created. These, together with the WUDS and PLDS, resulted in the ten advection-diffusion schemes tested in this work, which are listed in **Table 1**. This table also presents the value expected for the asymptotic order or order of accuracy (*p _{0}*) of the discretization error of the solution of T. These orders were obtained through

Acronym | Advection | Diffusion | Asymptotic order p_{0} |
---|---|---|---|

UDS1 | UDS-1 | CDS-2 | 1 |

Alpha | Alpha | CDS-2 | 1 |

CDS | CDS-2 | CDS-2 | 2 |

UDS2 | UDS-2 | CDS-2 | 2 |

WUDS | WUDS | WUDS | 2 |

PLDS | PLDS | PLDS | 2 |

ADS | ADS | CDS-2 | 2 |

TVD | TVD | CDS-2 | 2 |

QC2 | QUICK | CDS-2 | 2 |

QC3 | QUICK | CDS-4 and others-3 | 3 |

**Table 1:** The 10 advection-diffusion schemes tested in Eq. (1).

**Numerical solutions with repeated Richardson extrapolation (RRE)**

The numerical solution without extrapolation is obtained as described above for a set of grids g=[1,G], where g=1 is the coarsest grid, i.e., the one with the largest h size of the control volumes in **Figure 1**, and g=G is the finest grid, i.e., the one with the smallest h size of the control volumes in **Figure 1**. For each variable of interest (Ø), its numerical solution in grid g with m Richardson extrapolations is given by [8,9]

(4)

Where* r=h _{g-1}/hg* is the grid refinement ratio, and the variable

The numerical solutions without and with extrapolation were obtained, respectively, using the computational programs Peclet 1Dp 2.2 and Richardson 3.1. Both these programs were implemented in Fortran 95 (Intel 9.1) language and quadruple precision (Real*16); they are available at the web site ftp://ftp.demec.ufpr.br/CFD/projetos/ cfd4/. The simulations were performed in a core of an Intel Core 2 Quad processor inserted in a computer with a clock speed of 2.4 GHz, 8 GB of RAM and Windows XP 64 bit operating system. For each of the ten schemes, solutions were obtained in G=15 grids with N=5, 15, 45, 135, 405, 1,215, 3,645, 10,935, 32,805, 98,415, 295,245, 885,735, 2,657,205,7,971,615 and 23,914,845 real volumes; hence, the grid refinement ratio is r=3. This ratio was used to enable *T _{c}* to be obtained in the various grids, without any numerical approximation besides those employed to solve Eq. (1). The Alpha coefficient=0.05 was used for the Alpha scheme. In the case of the UDS2, ADS, TVD, QC2 and QC3 schemes, the solution is iterative. For these schemes, the iterative process was performed until the machine round-off error was reached, aiming to reduce the impact of the iteration error on the numerical error as much as possible. In all the calculations, for the 10 schemes and all the variables of interest, quadruple precision (Real*16) was used, aiming to reduce to a minimum the impact of the round-off error on the numerical error. As shown in [8,9], with double precision (Real*8) and RRE, the numerical error is already affected by the round-off error on grids with more than 100 nodes in each spatial direction. In addition, several tests were performed to reduce to a minimum the possible occurrence of a programming error. With these precautions, the main source of numerical error [19] is the discretization error, which can then by measured by means of [20]

E(Ø)=Ø -Φ (5)

where Ø and Φ represent, respectively, the numerical and exact analytical solutions of each variable of interest. To measure the numerical error with Eq. (5), the analytical solution (Φ) of each variable of interest was obtained using Maple software with 64 digits. For the ten schemes on the same grid (N), the ratio between the largest and the smallest computational memory needed to solve the problem varied from 1.28 to 1.12, respectively, for the coarsest and the finest grid. Therefore, the effect of the schemes on the memory is not very relevant and decreases as the grid becomes more refined. As for the CPU time required to solve the problem, the effect can be significant. Among the five non-iterative schemes (UDS1, Alpha, CDS, WUDS and PLDS), the ratio between the longest and the shortest CPU time needed to solve the problem is on average 1.21, considering the fifteen grids used. However, among the five iterative schemes (UDS2, ADS, TVD, QC2 and QC3), on the same grid (N) and for the same convergence criterion, the ratio between the shortest CPU time of the iterative schemes and the shortest CPU time of the non-iterative schemes is equal to at least 1.87 and at most 4.35, depending on N. And the ratio between the longest CPU time of the iterative schemes and the shortest CPU time of the non-iterative ones is equal to at least 6.90 and at most 19.7, depending on N. Due to space restrictions, the sections below present only a summary of the results. However, additional results can be seen in [11], and all the results of this work are described in [12]. This reference contains results of orders of accuracy and errors with and without RRE for all the variables, schemes and *Pe*, as well as for *Pe*=100.

**True orders of the discretization error**

**Table 2** presents the true orders (*p _{T}*) [19] of the discretization error of the four variables of interest and ten schemes in

T e _{c}L |
Tm |
q |
p_{T} |
---|---|---|---|

UDS1, Alpha | UDS1, Alpha | UDS1, CDS, UDS2, WUDS, PLDS, ADS, TVD | 1, 2, 3, ... |

UDS2, WUDS, PLDS, ADS, QC2 | UDS2, WUDS, PLDS, ADS, QC2, QC3 | QC2, QC3 | 2, 3, 4, ... |

CDS, TVD | CDS, TVD | --- | 2, 4, 6, ... |

QC3 | --- | --- | 3, 4, 5, ... |

**Table 2:** True orders (*p _{T}*) of the discretization error of the variables of interest.

**Errors and their primary variable orders**

**Figure 3** shows the error modulus (*E*) of the numerical solution of the variable *T _{c}* vs.

• The UDS1 and Alpha schemes are first-order accurate.

• The UDS2, CDS, WUDS, PLDS, ADS, TVD and QC2 schemes are second-order accurate.

• The QC3 scheme is third-order accurate.

• The three results above were expected according to their values of *p _{0}* in

• For the same *h* and *h *≤2.2x10^{-2}, i.e., with *N*≥45 nodes, the UDS1 and QC3 schemes have, respectively, the largest and the smallest error among the ten schemes.

• None of the schemes reach the minimum error of quadruple precision, even at the lowest *h*=4.2x10^{-8} or *N*=23,914,845 nodes. Therefore, the results of *Eh* of the ten schemes for *T _{c}* are in accordance with the known literature. This increases the reliability of the results obtained with RRE, according to the above results. As for the results of the ten schemes with extrapolation (

• For 3.4x10^{-6} ≤ h ≤ 3.0x10^{-5}, i.e., with *N* between 32,805 and 295,245 nodes, all the ten schemes reach the minimum level of error of quadruple precision. For lower values of *h*, the round-off error becomes larger than the discretization error, which results in an increase in the numerical error with the reduction of *h*.

• For the same *h* and *h* ≤ 7.4x10^{-3}, i.e., with N ≥ 135 nodes, the UDS1 and CDS schemes have, respectively, the largest and the smallest error among the ten schemes, while the discretization error prevails over the numerical error.

• For the same *h* and *h* ≤ 8.2x10^{-4}, i.e., with N ≥ 1,215 nodes, the TVD scheme has the second smallest error among the ten schemes, while the discretization error prevails over the numerical error.

• Except for the UDS1 and CDS schemes, the other eight schemes generally present crossovers among the various *Em* vs.* h* curves.

Here we have an unexpected result that is unknown in the literature: except in the first extrapolation (whose *h* ≈ 6.7x10^{-2} and *N*=15 nodes in **Figure 3**), the second-order CDS scheme presents a smaller error than the third-order QC3 scheme, when both use RRE type extrapolation. The probable explanation for this is that, already in the second extrapolation (whose *h* ≈ 2.2x10^{-2} and N=45 nodes in **Figure 3**), the theoretical order of the extrapolation of the CDS scheme is already six, while that of the QC3 scheme is five. As can be seen in **Table 2**, at each new extrapolation, the order of the CDS scheme is augmented by two units while that of the QC3 scheme increases by only one unit, increasing the difference between the error curves at each additional extrapolation. Similarly, for *h* ≤ 8.2x10^{-4}, the second-order TVD scheme presents a smaller error than the third-order QC3 scheme, when both use RRE. A comparison of the *Eh* and *Em* curves for *T _{c}* in

Error level for Eh and Em |
1E-6 | 1E-10 | 1E-14 |
---|---|---|---|

N of grid for the error level without RRE (Eh) |
1,215 | 98,415 | 7,971,615 |

N of grid for the error level with RRE (Em) |
45 | 135 | 405 |

Number of extrapolations (m) for Em |
2 | 3 | 4 |

N of Eh / N of Em |
27 | 729 | 19,683 |

**Table 3:** Grids required for three specific *T _{c}* errors with CDS and

N of grid |
45 | 1,215 | 32,805 |
---|---|---|---|

Error without RRE (|Eh|) for N |
3.1E-4 | 4.2E-7 | 5.8E-10 |

Error with RRE (|Em|) for N |
1.3E-7 | 1.6E-21 | 9.3E-30 |

Number of extrapolations (m) for Em |
2 | 5 | 8 |

|Eh| / |Em| |
2.4E3 | 2.6E14 | 6.2E19 |

**Table 4:** *Tc* errors with CDS and *Pe*=5 on three specific grids.

**Errors of the secondary variables**

**Figure 5** shows the error modulus (E) of the numerical solution of variable *Tm* vs. *h* for *Pe=5*, without (*Eh*) and with (*Em*) RRE. With regard to the results of the ten schemes without extrapolation (*Eh*) of *Tm*, it was found that:

• The UDS1 and Alpha schemes are first-order accurate.

• The UDS2, CDS, WUDS, PLDS, ADS, TVD and QC2 schemes are second-order accurate.

• The order of accuracy of the QC3 scheme degenerated from third to second-order because the error of the rectangle type numerical integration is second-order accurate.

• For the same *h *and *h* ≤2.2x10^{-2}, i.e., with *N* ≥45 nodes, the UDS1 and QC2 schemes have, respectively, the largest and the smallest error among the ten schemes.

• No scheme reached the minimum error level of quadruple precision, even at the lowest *h*=4.2x10^{-8} or *N*=23,914,845 nodes.

Therefore, the results of *Eh* of the ten schemes for *Tm* are in accordance with the known literature, considering the *p _{0}* (

• For 3.4x10^{-6} ≤*h* ≤3.0x10^{-5}, i.e., with N between 32,805 and 295,245 nodes, all the ten schemes reach the minimum error level of quadruple precision. For lower values of h, the round-off error becomes larger than the discretization error, which results in an increase of the numerical error with the reduction of h.

• For the same *h* and *h *≤ 2.2x10^{-2}, i.e., with N ≥45 nodes, the UDS1 scheme has the largest error among the ten schemes, while the discretization error prevails over the numerical error.

• For the same *h* and *h* ≤ 8.2x10^{-4}, i.e., with N ≥ 1,215 nodes, the TVD scheme has the second smallest error among the ten schemes, while the discretization error prevails over the numerical error.

• For the same h and h ≤ 6.7x10^{-2}, i.e., with N ≥ 15 nodes, the CDS scheme has the smallest error among the ten schemes, while the discretization error prevails over the numerical error.

• Except for the UDS1 and CDS schemes, the other eight schemes generally present crossovers between the various *Em* vs. *h* curves.

The probable explanation for the fact that the smallest errors of *Tm* are those of the TVD and CDS schemes is that these are the only ones that have values of pT=2, 4, 6, ... while the other six second-order schemes present *p _{T}*=2, 3, 4, ..., as indicated in

• The UDS1 scheme is first-order accurate.

• The order of accuracy of the UDS2, CDS, WUDS, PLDS, ADS and TVD schemes degenerated from second to first-order due to the UDS-2 approximation error used in the calculation of the first-order derivative of Eq. (2). For the same reason, the order of accuracy of the QC3 scheme degenerated from third to second-order. These degenerations in order are shown in [21] for this and other approximations, as well as for the Poisson, advection-diffusion and Burgers equations.

• The QC2 scheme maintained its second-order accuracy.

• For the same *h* and *h* ≤ 6.7x10-2, i.e., with *N* ≥ 15 nodes, the errors of the UDS2, CDS, WUDS, PLDS, ADS and TVD schemes are practically the same; the UDS1 scheme has a slightly smaller error than they do; the error of the QC3 scheme is smaller than that of UDS1; and lastly, the QC2 scheme has the smallest error among the nine schemes.

• None of the schemes reached the minimum error level of quadruple precision, even at the lowest *h*=4.2x10^{-8} or N=23,914,845 nodes.

**Table 2** lists the values of *p _{0}* and

• For 3.4x10^{-6} ≤ h ≤ 1.0x10^{-5}, i.e., with N between 98,415 and 295,245 nodes, all the schemes reach the minimum error level of quadruple precision. For lower values of *h*, the round-off error becomes larger than the discretization error, resulting in an increase of the numerical error with the reduction of* h*.

• For the same *h* and *h* ≤ 2.2x10^{-2}, i.e., with *N* ≥ 45 nodes, the UDS1 scheme has the largest error among the nine schemes, while the discretization error prevails over the numerical error.

• For the same* h* and *h* ≤ 2.2x10^{-2}, i.e., with *N* ≥ 45 nodes, the QC3 scheme has the second smallest error among the nine schemes, while the discretization error prevails over the numerical error.

• For the same *h *and *h* ≤ 6.7x10^{-2}, i.e., with *N* ≥ 15 nodes, the QC2 scheme has the smallest error among the nine schemes, while the discretization error prevails over the numerical error.

• For the other six schemes, there is no standard or pattern, since crossovers occur among the various *Em* vs. *h* curves. The probable explanation for the fact that the smallest errors of *q* are those of the QC3 and QC2 schemes is that these two are the only ones that have values of *p _{T}*=2,3,4, ... while the other seven schemes present

**Peclet number effect**

**Figure 7** shows the error modulus (*E*) vs. *h* of the numerical solution of variable *T _{c}* obtained with CDS for

• For the same* h* and *h* ≤ 2.0x10^{-1}, i.e., with N≥ 5 nodes, the higher the *Pe* the higher the *Eh*.

• The minimum error level of quadruple precision is reached for no *Pe*, even at the lowest *h*=4.2x10^{-8} or *N*=23,914,845 nodes. Therefore, these results of *Eh* are in line with the known literature. This increases the reliability of the results obtained with RRE, based on the above results. With regard to the results with extrapolation (*Em*) of *T _{c}* shown in

• For the same *h* and *h* ≤ 6.7x10^{-2}, i.e., with N ≥ 15 nodes, the higher the *Pe* the higher the Em.

• For 9.1x10^{-5} ≤ h ≤ 7.4x10^{-3}, i.e., with N between 135 and 10,935 nodes, *Em* reaches the minimum error level of quadruple precision at the four values of *Pe*. For lower values of *h*, the round-off error becomes larger than the discretization error, causing the numerical error to increase with the reduction of *h*.

• For *Pe*=0.01, 0.1, 1 and 10, respectively, the minimum value of the error is reached practically at h=7.4x10^{-3}, 2.5x10^{-3}, 8.2x10^{-4} and 9.1x10^{-5}, i.e., for N=135, 405, 1,215 and 10,935 nodes, respectively. Therefore, the higher the *Pe* the higher the *N* (or the lower the *h*) at which the minimum level of Em is reached.

A comparison of the *Eh* and *Em* curves for *T _{c}* in

In this work, the Repeated Richardson Extrapolation (RRE) technique was tested to reduce the discretization error of the solution of the 1D advection-diffusion equation. RRE was used on three variables of interest whose numerical solutions were obtained with the ten advection-diffusion schemes described in **Table 1**. Based on this work, it was found that:

a) RRE is extremely efficient in reducing the discretization error of primary and secondary variables whose solutions were obtained with schemes of first, second and third orders of accuracy, and five values of *Pe*, as indicated in **Figures 3, 5, 6 and 7**.

b) The second-order CDS scheme is the one that presented the best performance with RRE for the primary variable. In other words, among the ten schemes and for the same grid, the CDS has the smallest error when associated with RRE, surpassing the third-order QC3 scheme, as shown in **Figure 3**.

c) For secondary variables, the best scheme depends on the combination of the *p _{T}* values of the primary variable and the

d) With RRE, the highest error reduction is obtained with some scheme that has the highest variation between two subsequent values of *p _{T}*, and not with the scheme that has the highest value of

e) Two schemes with the same *p _{0}* can result in very different discretization errors when associated with RRE; for example, QC2 and CDS in

f) The error reduction resulting from the use of RRE is greater the higher the number of extrapolations; for example, **Figure 3**.

g) The round-off error limits the reduction of the numerical error when using RRE. This difficulty is minimized by using quadruple precision in the calculations; for example, **Figure 3**.

h) The higher the Peclet number the higher the value of the error for a given grid, without or with RRE. However, RRE is efficient in reducing the error at all the tested Peclet values; as indicated in **Figures 3 and 7**.

i) With CDS and RRE, the solution of 1D advection-diffusion for *Pe*=1 resulted in a reduction of the error equivalent to the solution of 2D diffusion with the same number of volumes in each direction. For *Pe*>1, the 1D advection-diffusion error is smaller than that of pure 2D diffusion; and for *Pe*<1, this reduction is greater.

j) Asymptotically, among various schemes, the ones that have the highest values of *p _{T}* are the ones that have the lowest values of error when associated with RRE; for example, CDS and TVD for

(i) Although the results of this work were obtained with the finite volume method, the same qualitative results should be obtained with other numerical methods, e.g., with the finite difference method;

(ii) The same qualitative results should be obtained for 2D and 3D advection-diffusion or more complex problems; and

(iii) The same qualitative results should be obtained with other schemes or numerical approximations.

The authors would like to acknowledge the financial support provided by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Níível Superior, Brazil), Fundação Araucária (Paraná, Brazil) and the Brazilian Space Agency (AEB), by the Uniespaço Program. The first author is supported by a CNPq scholarship. The authors would also like to acknowledge the suggestions provided by the reviewers.

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