Anunciacao et al 2019a - Revision history

Rimni at 11:18, 12 February 2021

2021-02-12T11:18:50Z

Rimni: /* 5.4 Algorithm */

2021-02-12T11:16:28Z

‎5.4 Algorithm

Rimni at 14:12, 18 December 2020

2020-12-18T14:12:55Z

Rimni: /* 5.3 Analysis of the velocities */

2020-12-18T12:30:48Z

‎5.3 Analysis of the velocities

Rimni at 12:29, 18 December 2020

2020-12-18T12:29:17Z

Rimni at 11:29, 18 December 2020

2020-12-18T11:29:52Z

Rimni at 11:27, 18 December 2020

2020-12-18T11:27:58Z

Show changes

Marcioalexandro at 00:03, 18 March 2020

2020-03-18T00:03:33Z

Rimni at 12:25, 10 March 2020

2020-03-10T12:25:02Z

Show changes

Rimni at 09:12, 10 February 2020

2020-02-10T09:12:11Z

@@ Line 1,250: / Line 1,250: @@
 For the ideal MG method, <math> p = 1</math>, meaning that the computational effort increases linearly with the size of the grid [3,6]. Thus, for a given hardware and compilator, the closer ''p'' is to the unity, the more efficient the algorithm is and the lower the value of ''c'', the faster it is.
-Table 4 presents the values of ''p'' and ''c'' for the curve adjustments obtained for the studied methods (data from Figure 15). In this table, it can be observed that the value of <math>p \approx 1</math> for PCG-MG-ILU, being <math>p \approx 2</math> for GS, as expected from literature [6]. Besides that, the lower value for ''c'' appears for PCG-MG-ILU.
+[[#tab-4|Table 4]] presents the values of ''p'' and ''c'' for the curve adjustments obtained for the studied methods (data from Figure 15). In this table, it can be observed that the value of <math>p \approx 1</math> for PCG-MG-ILU, being <math>p \approx 2</math> for GS, as expected from literature [6]. Besides that, the lower value for ''c'' appears for PCG-MG-ILU.
 <div class="center" style="font-size: 75%;"> '''Table 4'''. Values of ''p'' and ''c'' for the algorithms of the studied methodologies</div>

@@ Line 1,177: / Line 1,177: @@
 <div id='img-13'></div>
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
 |style="padding:10px;"| [[Image:Draft_Anunciacao_160657346-image135.jpeg|456px]]
@@ Line 1,212: / Line 1,212: @@
 </div>
-<div id='tab-1'></div>
+<div id='tab-3'></div>
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
@@ Line 1,254: / Line 1,254: @@
 <div class="center" style="font-size: 75%;"> '''Table 4'''. Values of ''p'' and ''c'' for the algorithms of the studied methodologies</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:35%;"
+<div id='tab-4'></div>
-|-
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Metodology'''
+|-style="text-align:center"
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|''c''
+! Metodology !! ''c'' !! ''p''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|''p''
 |-
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|GS
+|  style="text-align: center;vertical-align: top;"|GS
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5.332E-04
+|  style="text-align: center;vertical-align: top;"|5.332E-04
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1.855
+|  style="text-align: center;vertical-align: top;"|1.855
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|PCG-MG-ILU
+|  style="text-align: center;vertical-align: top;"|PCG-MG-ILU
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|4.272E-05
+|  style="text-align: center;vertical-align: top;"|4.272E-05
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.984
+|  style="text-align: center;vertical-align: top;"|0.984
 |}

@@ Line 1,170: / Line 1,170: @@
 ===5.4 Algorithm===
-After the tests of the two previous phases (sections 5.2 and 5.3), whose objective was to identify the best methods for the solution of the equations for the variables ''p'' (Poisson equation) and ''u'' and ''v'' (Reaction-Diffusion equations), we get to the point where the best methods were applied simultaneously to solve the Navier-Stokes equations.
+After the tests of the two previous phases (sections 5.2 and 5.3), whose objective was to identify the best methods for the solution of the equations for the variables <math>p</math> (Poisson equation) and <math>u</math> and <math>v</math> (Reaction-Diffusion equations), we get to the point where the best methods were applied simultaneously to solve the Navier-Stokes equations.
 The results presented below are compared the classic Gauss-Seidel (GS) method in its singlegrid version (only grid method) so that the gains can be measured when using the proposed algorithm (adaptation of the Projection method presented in Algorithm 1), that is, using the PCG-MG-ILU solver in the solution of the linear systems given in steps 1 and 2 of Algorithm 1.
-Figure 13 presents the behavior of <math>l_2</math>-norm of the nondimensionalized residue by the initial estimate in function of the number of iterations for the problem with <math>N = 128\times128</math> volumes for the variables ''u'', ''v'' and ''p''. Based on this figure, it can be noticed that the PCG-MG-ILU method reached the stop criterion ε = 10<sup>–10</sup> in an extremely lower number of iterations, both for ''p ''and for ''u'' and ''v''.
+[[#img-13|Figure 13]] presents the behavior of <math>l_2</math>-norm of the nondimensionalized residue by the initial estimate in function of the number of iterations for the problem with <math>N = 128\times128</math> volumes for the variables <math>u</math>, <math>v</math> and <math>p</math>. Based on this figure, it can be noticed that the PCG-MG-ILU method reached the stop criterion ε = 10<sup>–10</sup> in an extremely lower number of iterations, both for <math>p</math> and for <math>u</math> and <math>v</math>.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
 |style="padding:10px;"| [[Image:Draft_Anunciacao_160657346-image135.jpeg|456px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 13'''. <math>l_2</math>-norm of the nondimensionalized residue by norm <math>l_2</math> for the initial
+| colspan="1" style="padding-bottom:10px;"| '''Figure 13'''. <math>l_2</math>-norm of the nondimensionalized residue by norm <math>l_2</math> for the initial estimate ''versus'' number of iterations for the problem with <math>N = 128\times128</math> for <math>u</math>, <math>v</math> and <math>p</math>
-−
-estimate ''versus'' number of iterations for the problem with <math>N = 128\times128</math> for ''u'', ''v'' and ''p''
 |}
-Figure 14 illustrates the behavior of the empirical convergence factor with the increase of the size of the problem. Based on this figure, it can be concluded that the PCG-MG-ILU method presented much better results for the convergence factors for the three variables of interest, comparing with the classic GS method.
+[[#img-14|Figure 14]] illustrates the behavior of the empirical convergence factor with the increase of the size of the problem. Based on this figure, it can be concluded that the PCG-MG-ILU method presented much better results for the convergence factors for the three variables of interest, comparing with the classic GS method.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
 |style="padding:10px;"|[[Image:Draft_Anunciacao_160657346-image136.jpeg|456px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 14'''. Empirical convergence factor (<math>q^{(k)}</math>) ''versus'' size of the problem (''N'') for ''u'', ''v'' and ''p''
+| colspan="1" style="padding-bottom:10px;"| '''Figure 14'''. Empirical convergence factor (<math>q^{(k)}</math>) ''versus'' size of the problem (''N'') for <math>u</math>, <math>v</math> and <math>p</math>
 |}
-Figure 15 contains the values of the CPU times (in seconds) of the simulations with the increase of the size of the problem. Based on this figure, it can be noticed that the PCG-MG-ILU methodology was the one that converged faster, being quite evident the difference in CPU time between the proposed methodology and the classic GS method.
+[[#img-15|Figure 15]] contains the values of the CPU times (in seconds) of the simulations with the increase of the size of the problem. Based on this figure, it can be noticed that the PCG-MG-ILU methodology was the one that converged faster, being quite evident the difference in CPU time between the proposed methodology and the classic GS method.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 1,206: / Line 1,207: @@
-Table 3 presents the Speed-up (Eq.(56)) on the GS method in relation to the PCG-MG-ILU method. According to the data on this table, the methodology that uses the PCG-MG-ILU method proved much more efficient than the methodology that utilizes the GS method, reaching a 105 times higher velocity for the biggest problem compared in this table. It can also be noticed that the Speed-up increases as the size of the problem increases, which is a highly desirable property.
+[[#tab-3|Table 3]] presents the Speed-up (Eq.(56)) on the GS method in relation to the PCG-MG-ILU method. According to the data on this table, the methodology that uses the PCG-MG-ILU method proved much more efficient than the methodology that utilizes the GS method, reaching a 105 times higher velocity for the biggest problem compared in this table. It can also be noticed that the Speed-up increases as the size of the problem increases, which is a highly desirable property.
 <div class="center" style="font-size: 75%;">'''Table 3'''. Values of GS Speed-up in relation to PCG-MG-ILU
 </div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:25%;"
+<div id='tab-1'></div>
-|-
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|''N''
+|-style="text-align:center"
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|''S<sub>P</sub>''
+!  style="text-align: center;"|''N'' !!   style="text-align: center;vertical-align: top;"|''S<sub>P</sub>''
 |-
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|8<math>\times</math>8
+|  style="text-align: center;vertical-align: top;"|8<math>\times</math>8
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|3.013
+|  style="text-align: center;vertical-align: top;"|3.013
 |-
 |  style="text-align: center;vertical-align: top;"|16<math>\times</math>16
@@ Line 1,228: / Line 1,229: @@
 |  style="text-align: center;vertical-align: top;"|31.181
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|128<math>\times</math>128
+|  style="text-align: center;vertical-align: top;"|128<math>\times</math>128
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|105.242
+|  style="text-align: center;vertical-align: top;"|105.242
 |}

@@ Line 1,085: / Line 1,085: @@
 |style="padding:10px;"| [[Image:Draft_Anunciacao_160657346-image131.jpeg|432px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 9'''. <math>l_\infty </math>-norm for the numerical error (<math>||E||_\infty</math>) ''versus'' size of the problem (''N'') for ''p''
+| colspan="1" style="padding:10px;"| '''Figure 9'''. <math>l_\infty </math>-norm for the numerical error (<math>||E||_\infty</math>) ''versus'' size of the problem (''N'') for ''p''
 |}
@@ Line 1,092: / Line 1,092: @@
 <div id='img-10'></div>
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 50%;"
 |-
 |style="padding:10px;"| [[Image:Draft_Anunciacao_160657346-image132.jpeg|456px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 10'''. <math>l_2</math>-norm for the nondimensionalized residue by <math>l_2</math>-norm of the initial estimate ''versus'' number of iterations for the problem with <math>N = 1024\times 1024</math> for ''u''
+| colspan="1" style="padding:10px;"| '''Figure 10'''. <math>l_2</math>-norm for the nondimensionalized residue by <math>l_2</math>-norm of the initial estimate ''versus'' number of iterations for the problem with <math>N = 1024\times 1024</math> for ''u''
 |}

@@ Line 1,078: / Line 1,078: @@
 In this phase, it is intended to determine the best method to solve the Reaction-Diffusion equations, which model the velocities ''u'' and ''v'', determining the best method for pressure presented in the previous section, which was PCG-MG-ILU. Since the results for the velocities ''u'' and ''v'' for all the parameters are similar, it was chose to display only the results for ''u''.
-Figure 9 shows the behavior of the numerical error ''versus'' the size of the problem, for the several solution methods. It can be noticed that the four studied methods presented an expected characteristic for the convergent iterative methods, that is, the decrease in the numerical error with the increase in the size of the problem. It can also be noticed that the presented error have values which are very close among themselves.
+[[#img-9|Figure 9]] shows the behavior of the numerical error ''versus'' the size of the problem, for the several solution methods. It can be noticed that the four studied methods presented an expected characteristic for the convergent iterative methods, that is, the decrease in the numerical error with the increase in the size of the problem. It can also be noticed that the presented error have values which are very close among themselves.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 1,088: / Line 1,089: @@
-Figure 10 displays the behavior of <math>_2</math>-norm of the nondimensionalized residue by <math>l_2</math>-norm of the initial estimate in function of the number of iterations for the problem with <math>N = 1024\times 1024</math> volumes. Based on this figure, it can be noticed that the PCG-MG-ILU and the MG-ILU methods were the ones which achieved the stop criterion in the lowest number of iterations.
+[[#img-10|Figure 10]] displays the behavior of <math>_2</math>-norm of the nondimensionalized residue by <math>l_2</math>-norm of the initial estimate in function of the number of iterations for the problem with <math>N = 1024\times 1024</math> volumes. Based on this figure, it can be noticed that the PCG-MG-ILU and the MG-ILU methods were the ones which achieved the stop criterion in the lowest number of iterations.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
 |style="padding:10px;"| [[Image:Draft_Anunciacao_160657346-image132.jpeg|456px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 10'''. <math>l_2</math>-norm for the nondimensionalized residue by <math>l_2</math>-norm of the
+| colspan="1" style="padding-bottom:10px;"| '''Figure 10'''. <math>l_2</math>-norm for the nondimensionalized residue by <math>l_2</math>-norm of the initial estimate ''versus'' number of iterations for the problem with <math>N = 1024\times 1024</math> for ''u''
-−
 initial estimate ''versus'' number of iterations for the problem with <math>N = 1024\times 1024</math> for ''u''
 |}
-Figure 11 illustrates the behavior of the empirical convergence factor in function of the size of the problem. Based on this figure, it can be concluded that the PCG-MG-ILU method presented the best results, since it generated the best values for the convergence factors (closest values to zero), besides decreasing its value with the increase of the size of the problem.
+[[#img-11|Figure 11]] illustrates the behavior of the empirical convergence factor in function of the size of the problem. Based on this figure, it can be concluded that the PCG-MG-ILU method presented the best results, since it generated the best values for the convergence factors (closest values to zero), besides decreasing its value with the increase of the size of the problem.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 1,110: / Line 1,111: @@
-Figure 12 shows the CPU times (in seconds) of the simulations with the increase of the size of the problem. Based on this figure, it can be noticed that the computational times were very close, with a slight advantage for the PCG-MG-ILU methodology, which can be confirmed in the results presented in the sequence.
+[[#img-12|Figure 12]] shows the CPU times (in seconds) of the simulations with the increase of the size of the problem. Based on this figure, it can be noticed that the computational times were very close, with a slight advantage for the PCG-MG-ILU methodology, which can be confirmed in the results presented in the sequence.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 1,120: / Line 1,122: @@
-Table 2 presents the Speed-up (Eq.(56)) of the MG-GS, MG-ILU and PCG-MG-GS methods in relation to the PCG-MG-ILU method. In this table, it can be confirmed that the CPU times are close among themselves, but there is an advantage with the use of PCG-MG-ILU, that is, the method that most accelerated the convergence of the iterative process for the presented problem.
+[[#tab-2|Table 2]] presents the Speed-up (Eq.(56)) of the MG-GS, MG-ILU and PCG-MG-GS methods in relation to the PCG-MG-ILU method. In this table, it can be confirmed that the CPU times are close among themselves, but there is an advantage with the use of PCG-MG-ILU, that is, the method that most accelerated the convergence of the iterative process for the presented problem.
 <div class="center" style="font-size: 75%;">'''Table 2'''. Values of Speed-up for MG-GS, MG-ILU and PCG-MG-GS in relation to PCG-MG-ILU
 </div>
-{| style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
+<div id='tab-2'></div>
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|''N''
+|  style="text-align: center;vertical-align: top;"|16<math>\times</math>16
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''MG-GS'''
+|  style="text-align: center;vertical-align: top;"|1
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''MG-ILU'''
+|  style="text-align: center;vertical-align: top;"|1
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''PCG-MG-GS'''
+|  style="text-align: center;vertical-align: top;"|1
-|-
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|16<math>\times</math>16
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1
 |-
 |  style="text-align: center;vertical-align: top;"|32<math>\times</math>32
@@ Line 1,162: / Line 1,162: @@
 |  style="text-align: center;vertical-align: top;"|1.524
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1024<math>\times</math>1024
+|  style="text-align: center;vertical-align: top;"|1024<math>\times</math>1024
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.337
+|  style="text-align: center;vertical-align: top;"|1.337
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.228
+|  style="text-align: center;vertical-align: top;"|1.228
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.535
+|  style="text-align: center;vertical-align: top;"|1.535
 |}

@@ Line 983: / Line 983: @@
 |style="padding:10px;"|[[Image:Draft_Anunciacao_160657346-image127.jpeg|456px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 6'''. <math>l_2</math>-norm for the nondimensionalized residue by the norm of initial estimate ''versus'' number of
+| colspan="1" style="padding:10px;"| '''Figure 6'''. <math>l_2</math>-norm for the nondimensionalized residue by the norm of initial estimate ''versus'' number of
 iterations for the problem with <math>N' = 1024\times 1024</math> for ''p''
 |}

@@ Line 1,277: / Line 1,277: @@
 [1] Mitin A.V. Linear extrapolation in an iterative method for solving systems of equations, U.S.S.R. Comput. Maths. Math. Phys., 25(2):1-6, 1985.
-[2] Fedorenko R.P. On the speed of convergence of an iteration process. USSR Comput. Math. And Math. Phys., 4(3):227-235, 1964.
+[2] Fedorenko R.P. On the speed of convergence of an iteration process. U.S.S.R. Comput. Math. And Math. Phys., 4(3):227-235, 1964.
 [3] Brandt A. Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation,  31:333-390, 1977.
-[4] Pletcher R., Tannehill J., Anderson D. Computational fluid mechanics and heat transfer.  Series in Computational and Physical processes in Mechanics and Thermal Sciences, Second Edition, Taylor & Francis, 1997.
+[4] Pletcher R., Tannehill J., Anderson D. Computational fluid mechanics and heat transfer.  Series in Computational and Physical processes in Mechanics and Thermal Sciences, 2nd ed., Taylor & Francis, 1997.
-[5] Briggs W.L., Henson V.E., McCormick S.F. A multigrid tutorial. SIAM, 2nd ed., 2000.
+[5] Briggs W.L., Henson V.E., McCormick S.F. A multigrid tutorial. 2nd ed., SIAM, 2000.
 [6] Trottenberg U., Oosterlee C., Schüller A. Multigrid. Academic Press, 2001.
@@ Line 1,289: / Line 1,289: @@
 [7] Saad Y. Iterative methods for sparse linear systems. SIAM, 2003.
-[8] Batchelor G. An introduction to fluid dynamics. Cambridge University Press, Cambridge, 1970.
+[8] Batchelor G. An introduction to fluid dynamics. Cambridge University Press, 1970.
 [9] Peyret R., Taylor T.D. Computational methods for fluids flow. Springer-Verlag, 1983.
@@ Line 1,295: / Line 1,295: @@
 [10] Panton R.L. Incompressible flow. John Wiley & Sons, 1984.
-[11] Fletcher C.A.J. Computational methods for fluid dynamics. Vol. 1, Springer, 2nd ed., 1992.
+[11] Fletcher C.A.J. Computational methods for fluid dynamics. Vol. 1, 2nd ed., Springer, 1992.
-[12] Maliska C.R. Transferência de calor e mecânica dos fluidos computacional. LTC, 2nd ed., 2004.
+[12] Maliska C.R. Transferência de calor e mecânica dos fluidos computacional. 2nd ed., LTC, 2004.
 [13] Ferziger J.H., Peric M. Computational methods for fluid dynamics. Springer-Verlag, 2002.
@@ Line 1,313: / Line 1,313: @@
 [19] Ernst O.G., Gander M.J. Why it is difficult to solve Helmholtz problems with classical iterative methods? Computational Science and Engineering, 83:325-363, 2012.
-[20] Incropera F.P., Dewitt D.P. Bergman T.L., Lavine, A.S. Fundamentals of heat and mass transfer.  John Wiley & Sons, 6 ed., 2007.
+[20] Incropera F.P., Dewitt D.P. Bergman T.L., Lavine, A.S. Fundamentals of heat and mass transfer.  6 ed., John Wiley & Sons, 2007.
 [21] Harlow F.H., Welch J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. The Physics of  Fluids, 8:2182-2189, 1965.
@@ Line 1,335: / Line 1,335: @@
 [30] Kalland K.A. Navier-stokes solver for single and two-phase flow. Master Thesis, University of Oslo, 2008.
-[31] Villar M.M. Análise numérica detalhada de escoamentos multifásicos bidimensionais. Thesis UFU, Uberlandia, MG, Brazil, 2007.
+[31] Villar M.M. Análise numérica detalhada de escoamentos multifásicos bidimensionais. Thesis UFU, Uberlandia, Brazil, 2007.
 [32] Wesseling P. An introduction to multigrid methods. John Wiley & Sons, 1992.

@@ Line 1,275: / Line 1,275: @@
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] Mitin, A. V. Linear extrapolation in an iterative method for solving systems of equations, U.S.S.R. Comput. Maths. Math. Phys., Vol. 25, 2:1-6, 1985.
+[1] Mitin, A. V. Linear extrapolation in an iterative method for solving systems of equations, U.S.S.R. Comput. Maths. Math. Phys., 25(2):1-6, 1985.
 [2] Fedorenko, R. P. On the Speed of Convergence of an Iteration Process, USSR Comput. Math. And Math. Phys., Vol. 4, 3, 1964.