Vassoler Rutz et al 2018a - Revision history

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Rimni: /* 4. Numerical results */

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‎4. Numerical results

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 <div class="auto" style="width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] Thekale T. Gradl, K. Klamroth, U. Rüde, U. Optimizing the number of multigrid cycles in the full multigrid algorithm. Numerical Linear Algebra with Applications. Vol. 17 (2010) 199–210.
+[1] Thekale T. Gradl, Klamroth K.,  Rüde U. Optimizing the number of multigrid cycles in the full multigrid algorithm. Numerical Linear Algebra with Applications, Vol. 17, 199–210,  2010.
-[2] R.H. Pletcher, J.C. Tannehill, D.A. Anderson, Computational Fluid Mechanics and Heat Transfer, 3ª ed. CRC Press, USA, 2013.
+[2] Pletcher R.H.,  Tannehill J.C.,  Anderson D.A. Computational fluid mechanics and heat Transfer. 3ª ed. CRC Press, USA, 2013.
 [3] G.H. Golub, J.M. Ortega, Scientific Computing and Differential Equations: an Introduction to Numerical Methods, Academic Press, 1992.
@@ Line 963: / Line 963: @@
 [7] P. Wesseling, An Introduction to Multigrid Methods, John Wiley & Sons, England, 1992.
-[8] F. Oliveira, M.A.V. Pinto, C.H. Marchi, L.K. Araki. Optimized partial semicoarsening multigrid algorithm for heat diffusion problems and anisotropic grids. Applied Mathematical Modelling. Vol. 36 (2012) 4665–4676.
+[8] F. Oliveira, M.A.V. Pinto, C.H. Marchi, L.K. Araki. Optimized partial semicoarsening multigrid algorithm for heat diffusion problems and anisotropic grids. Applied Mathematical Modelling, Vol. 36 (2012) 4665–4676.
-[9] J.A. Rabi, M.J.S. de Lemos. Optimization of convergence acceleration in multigrid numerical solutions of conductive-convective problems. Applied Mathematics and Computational. Vol. 124 (2001) 215-226.
+[9] J.A. Rabi, M.J.S. de Lemos. Optimization of convergence acceleration in multigrid numerical solutions of conductive-convective problems. Applied Mathematics and Computational, Vol. 124 (2001) 215-226.
 [10] R. Wienands, W. Joppich, Practical Fourier Analysis for Multigrid Methods, CRC Press, USA, 2005.
@@ Line 971: / Line 971: @@
 [11] K. Johannsen. A robust 9-point ILU smoother for anisotropic problems, IWR Preprint, University of Heidelberg, 2005.
-[12] S.A. Vinogradova, L.A. Krukier, The use of incomplete LU decomposition in modeling convection-diffusion processes in an anisotropic medium. Mathematical Models and Computer Simulations. Vol. 5 (2013) 190-197.
+[12] S.A. Vinogradova, L.A. Krukier, The use of incomplete LU decomposition in modeling convection-diffusion processes in an anisotropic medium. Mathematical Models and Computer Simulations, Vol. 5 (2013) 190-197.
 [13] G. Vassoler-Rutz, M.A.V. Pinto, R. Suero, Comparison of the physical anisotropy of multigrid method for two-dimensional diffusion equation, in: Proceedings of COBEM, Rio de Janeiro, Brazil, (2015) 6-11.
-[14] M.A. V. Pinto, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee. On the robustness of ILU smoothers on triangular grids, Applied Numerical Mathematics. Vol. 106 (2016), 37-52.
+[14] M.A. V. Pinto, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee. On the robustness of ILU smoothers on triangular grids, Applied Numerical Mathematics, Vol. 106 (2016), 37-52.
-[15] S.R. Franco, F.J. Gaspar, M.A.V. Pinto, C. Rodrigo. Multigrid method based on a space-time approach with standard coarsening for parabolic problems. Applied Mathematics and Computation. Vol. 317 (2018) 25-34.
+[15] S.R. Franco, F.J. Gaspar, M.A.V. Pinto, C. Rodrigo. Multigrid method based on a space-time approach with standard coarsening for parabolic problems. Applied Mathematics and Computation, Vol. 317 (2018) 25-34.
-[16] M.L. Oliveira, M.A. V. Pinto, S.F.T. Gonçalves, G. Vassoler-Rutz. On the Robustness of the xy-Zebra-Gauss-Seidel Smoother on an Anisotropic Diffusion Problem. CMES. Vol.117 (2018) 251-270.
+[16] M.L. Oliveira, M.A. V. Pinto, S.F.T. Gonçalves, G. Vassoler-Rutz. On the Robustness of the xy-Zebra-Gauss-Seidel Smoother on an Anisotropic Diffusion Problem. CMES., Vol.117 (2018) 251-270.
 [17] F. Oliveira. Effect of Two-Dimensional Anisotropic Meshes on the Performance of the Geometric Multigrid Method. Tese de doutorado, UFPR, 2010. ''In portuguese''.
-[18] P. Wesseling, C.W. Oosterlee. Geometric Multigrid with Applications to Computational Fluid Dynamics. Journal of Computation and Applied Mathematics. Vol. 128 (2001) 311-334.
+[18] P. Wesseling, C.W. Oosterlee. Geometric Multigrid with Applications to Computational Fluid Dynamics. Journal of Computation and Applied Mathematics, Vol. 128 (2001) 311-334.
-[19] J.L. Guermond, P. Minev, Jie Shen. An overview of prejection methods for incompressible flows. Computer methods in apllied mechanics and engineering. Vol. 195 (2006)  6011-6045.
+[19] J.L. Guermond, P. Minev, Jie Shen. An overview of prejection methods for incompressible flows. Computer Methods in Apllied Mechanics and Engineering, Vol. 195 (2006)  6011-6045.
 [20] R.L. Burden, J.D. Faires, Numerical Analysis, 9ª ed., Cengage Learning, USA, 2010.
 </div>

@@ Line 814: / Line 814: @@
 In order to assess the performance of the multigrid method with different anisotropy coefficients, a curve adjustment of the form <math>{{t}_{CPU}}=c{{N}^{p}}</math> [20] was made, where <math>p</math> represents the complexity order of the solver, ''N'' is the number of grid points and<math>N</math> <math>c</math> is a constant that depends on the method. The closer the value of <math>p</math> is to one, the better the performance of the method used. Ideally, multigrid presents <math>p=1</math>, what means that the CPU time grows linearly with the increase of ''N''. Results are shown in [[#tab-2|Table 2]] for both restrictions assessed<math>N</math> (<math>\mbox{FW}</math> and  <math display="inline">{\mbox{PW}}_\mbox{x}</math>).
-One can observe in [[#tab-2|Table 2]] that, for every orthotropy employed, the multigrid method has a good performance, since <math>p\approx1</math> in every case. These results prove the efficiency and robustness of Algorithm 1, proposed in this work.
+One can observe in [[#tab-2|Table 2]] that, for every orthotropy employed, the multigrid method has a good performance, since <math>p\approx1</math> in every case. These results prove the efficiency and robustness of [[#algorithm-1|Algorithm 1]], proposed in this work.
 <div class="center" style="font-size: 75%;">
@@ Line 932: / Line 932: @@
 - With bilinear interpolation, <math display="inline">{\mbox{ILU}}_{\mbox{EN}}</math> for <math>0<\varepsilon <<1</math> and <math>{\mbox{ILU}}_{\mbox{NE}}</math> for  <math>\varepsilon>>1</math>, the lowest values of  <math>{{\rho }_{loc}}</math> are obtained with <math>\mbox{FW}</math> and <math display="inline">{\mbox{PW}}_\mbox{x}</math> restriction operators, and the values of  <math>{{\rho }_{loc}}</math> with these operators are very similar.
-- Using Algorithm 1, <math>{{\rho }_{loc}}\approx{{\rho }_{h}}<<1</math>  for every orthotropy coefficient assessed and <math>{{\rho }_{h}}\to {{\rho }_{loc}}</math> as the grid becomes more refined.
+- Using [[#algorithm-1|Algorithm 1]], <math>{{\rho }_{loc}}\approx{{\rho }_{h}}<<1</math>  for every orthotropy coefficient assessed and <math>{{\rho }_{h}}\to {{\rho }_{loc}}</math> as the grid becomes more refined.
 - The  <math display="inline">t_{CPU}\left(\mbox{FW}\right)\approx t_{CPU}\left({\mbox{PW}}_\mbox{x}\right)</math> for every value of <math>\varepsilon</math> assessed.
-- The complexity order <math>p</math> of the multigrid method with Algorithm 1 is close to one for every orthotropy assessed. For <math>\varepsilon ={{10}^{-4}}</math>, for example, <math>p=1.07747</math> with this algorithm.
+- The complexity order <math>p</math> of the multigrid method with [[#algorithm-1|Algorithm 1]] is close to one for every orthotropy assessed. For <math>\varepsilon ={{10}^{-4}}</math>, for example, <math>p=1.07747</math> with this algorithm.
-- The computational cost of multigrid depends on the number of flops of the restriction in a V-cycle. Using Algorithm 1 with  <math display="inline">{\mbox{PW}}_\mbox{x}</math> restriction, the computational cost is 75% lower than with <math>\mbox{FW}</math> restriction.
+- The computational cost of multigrid depends on the number of flops of the restriction in a V-cycle. Using [[#algorithm-1|Algorithm 1]] with  <math display="inline">{\mbox{PW}}_\mbox{x}</math> restriction, the computational cost is 75% lower than with <math>\mbox{FW}</math> restriction.
-- The Algorithm 1 with  <math display="inline">{\mbox{PW}}_\mbox{x}</math> restriction proposed in this work is efficient, robust and has low computational cost.
+- The [[#algorithm-1|Algorithm 1]] with  <math display="inline">{\mbox{PW}}_\mbox{x}</math> restriction proposed in this work is efficient, robust and has low computational cost.
 == Acknowledgements ==
@@ Line 949: / Line 949: @@
 <div class="auto" style="width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] Thekale, T. Gradl, K. Klamroth, U. Rüde, U. Optimizing the number of multigrid cycles in the full multigrid algorithm. Numerical Linear Algebra with Applications. Vol. 17 (2010) 199–210.
+[1] Thekale T. Gradl, K. Klamroth, U. Rüde, U. Optimizing the number of multigrid cycles in the full multigrid algorithm. Numerical Linear Algebra with Applications. Vol. 17 (2010) 199–210.
 [2] R.H. Pletcher, J.C. Tannehill, D.A. Anderson, Computational Fluid Mechanics and Heat Transfer, 3ª ed. CRC Press, USA, 2013.

@@ Line 820: / Line 820: @@
 <div id='tab-2'></div>
-{| style="width: 76%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
+{| style="width: 66%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-
 | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;width: 25%;" |<math>\varepsilon</math>
@@ Line 830: / Line 830: @@
 | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;" |<math>1.07733</math>
 |-
-| style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;" |<span style="text-align: center; font-size: 75%;"> </span><math>10^{-2}</math>
+| style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;" |<math>10^{-2}</math>
 | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;" |<math>1.05023</math>
 | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;" |<math>1.05894</math>

@@ Line 41: / Line 41: @@
 Several works on the implementation of the multigrid method found in the literature demonstrate that the choice of the multigrid components is crucial for the convergence or not of the method. Trottenberg et al. [6] state that this choice is difficult and thus small changes can considerably improve convergence. In this sense, LFA can help this choice as it allows to predict the performance of the multigrid method, since it provides estimates of the convergence rates based on the variation of the multigrid method components.
-Pinto et. al. [14] solved an anisotropic diffusion problem using ILU in triangular grids. They used LFA to highlight the good smoothing properties of the solver and the asymptotic convergence of the multigrid.
+Pinto et al. [14] solved an anisotropic diffusion problem using ILU in triangular grids. They used LFA to highlight the good smoothing properties of the solver and the asymptotic convergence of the multigrid.
-Franco et. al. [15] performed LFA in transient problems and obtained the critical value of the parameter that represents the level of space-time anisotropy for 1D and 2D Fourier equations.
+Franco et al. [15] performed LFA in transient problems and obtained the critical value of the parameter that represents the level of space-time anisotropy for 1D and 2D Fourier equations.
-Oliveria et. al. [16] solved an 2D anisotropic diffusion equation. The equation was discretized by the Finite Difference Method (FDM) and Central Differencing Scheme (CDS). Correction Scheme (CS). An xy-zebra-GS smoother was proposed, which proved to be efficient and robust for the different anisotropy coefficients. They concluded that, the convergence factors calculated empirically and by LFA are in agreement.
+Oliveria et al. [16] solved an 2D anisotropic diffusion equation. The equation was discretized by the Finite Difference Method (FDM) and Central Differencing Scheme (CDS). Correction Scheme (CS). An xy-zebra-GS smoother was proposed, which proved to be efficient and robust for the different anisotropy coefficients. They concluded that, the convergence factors calculated empirically and by LFA are in agreement.
 A particular case of anisotropy is denominated orthotropy, which happens when the anisotropy occurs in orthogonal directions. In this work, an efficient and robust method for solving physical orthotropy problems using LFA is proposed. A two-dimensional diffusion mathematical model is considered, in which physical orthotropy appears in the coefficients and it will be denominated diffusion orthotropy. Equations were discretized using FDM with second-order central difference scheme.
@@ Line 145: / Line 145: @@
 where <math>T</math> is the unknown of the system.
-[[#img-1|Figure 1]](b) depicts the notation of the grid points in [[#img-1|Figure 1]](a). The points P (central), W (west), E (east), N (north) and S (south) in [[#img-1|Figure 1]](b) correspond to the points <math display="inline">(i,j),(i-1,j),(i+1,j),(i,j+1),(i,j-1)</math> in Figure 1(a), respectively.
+[[#img-1|Figure 1]](b) depicts the notation of the grid points in [[#img-1|Figure 1]](a). The points P (central), W (west), E (east), N (north) and S (south) in [[#img-1|Figure 1]](b) correspond to the points <math display="inline">(i,j),(i-1,j),(i+1,j),(i,j+1),(i,j-1)</math> in [[#img-1|Figure 1]](a), respectively.
 <div id='img-1'></div>
@@ Line 720: / Line 720: @@
-'''Remark 1:''' Pinto et. al. [14] solved an anisotropic diffusion problem using ILU in triangular grids and several anisotropies not aligned with ''x'' or ''y''. The authors noted that ILU<sub>NE</sub> and ILU<sub>EN</sub> were efficient for some of the anisotropies, making it is possible to adapt this algorithm to an alternating form so it will work well for any anisotropy.
+'''Remark 1:''' Pinto et al. [14] solved an anisotropic diffusion problem using ILU in triangular grids and several anisotropies not aligned with ''x'' or ''y''. The authors noted that ILU<sub>NE</sub> and ILU<sub>EN</sub> were efficient for some of the anisotropies, making it is possible to adapt this algorithm to an alternating form so it will work well for any anisotropy.
 '''Remark 2:'''   One of the biggest problems in the literature is the numerical resolution of the Navier-Stokes equation. Depending on its numerical formulation (simplec, projections [19]), a great computational effort is required at the numerical solution of the continuity equation, which can be represented by Poisson’s equation. Moreover, the algorithm depends on the mesh sweep and is independent of the complexity of the proposed problem equation. Therefore, this algorithm can be adapted to certain orthotropic problems with a certain degree of complexity.
@@ Line 848: / Line 848: @@
-The values obtained for the convergence factor, presented in section 4.1, and for the complexity order concerning to the <math>\text{PW}_x</math> and <math>\mbox{FW}</math> restriction operators (Table 2) are quite similar and thus insufficient to decide which one results in a more efficient algorithm.
+The values obtained for the convergence factor, presented in section 4.1, and for the complexity order concerning to the <math>\text{PW}_x</math> and <math>\mbox{FW}</math> restriction operators ([[#tab-2|Table 2]]) are quite similar and thus insufficient to decide which one results in a more efficient algorithm.
 To complement the analysis, the number of arithmetical operations performed in each V-cycle and at the restriction step for each one of the operators was assessed.

@@ Line 708: / Line 708: @@
 '''Algorithm 1.'''<br />
 _______________________________________________________
 if <math>\varepsilon >1</math> then
 :Apply  <math display="inline">{\mbox{ILU}}_{\mbox{NE}}</math> smoothing with '''REST '''restriction

@@ Line 711: / Line 711: @@
 if <math>\varepsilon >1</math> then
-Apply  <math display="inline">{\mbox{ILU}}_{\mbox{NE}}</math> smoothing with '''REST '''restriction
+:Apply  <math display="inline">{\mbox{ILU}}_{\mbox{NE}}</math> smoothing with '''REST '''restriction
-else if <math>\varepsilon =1</math> then
+::else if <math>\varepsilon =1</math> then
-Apply  <math display="inline">{\mbox{ILU}}_{\mbox{NE}}</math> smoothing with <math>\mbox{FW}</math> restriction
+:Apply  <math display="inline">{\mbox{ILU}}_{\mbox{NE}}</math> smoothing with <math>\mbox{FW}</math> restriction
-else
+::else
-Apply  <math display="inline">{\mbox{ILU}}_{\mbox{EN}}</math> '''REST '''restriction
+:Apply  <math display="inline">{\mbox{ILU}}_{\mbox{EN}}</math> '''REST '''restriction
 end if<br />

@@ Line 744: / Line 744: @@
 It is observed that <math>\rho_{loc} \approx \rho_{h} <<1</math>  for every orthotropy coefficients assessed. Furthermore, <math>{{\rho }_{loc}}</math> calculated by LFA is in accordance with <math>\rho_h</math> calculated experimentally.
-[[#img-7|Figure 7]] depicts the numerical asymptotic convergence factor <math>{{\rho }_{loc}}</math> calculated by LFA and the experimental asymptotic convergence factor <math>\rho_h</math>, for different orthotropy coefficients for different grids. Algorithm 1 with  '''REST''' = <math display="inline">{\mbox{PW}}_\mbox{x}</math>, <math>\nu=2</math> and bilinear prolongation was used. As the grid  becomes more refined, <math>\rho_{h} \rightarrow\rho_{loc} </math> for every orthotropy coefficient analyzed, what demonstrates the robustness of the methodology assessed.
+[[#img-7|Figure 7]] depicts the numerical asymptotic convergence factor <math>{{\rho }_{loc}}</math> calculated by LFA and the experimental asymptotic convergence factor <math>\rho_h</math>, for different orthotropy coefficients for different grids. [[#algorithm-1|Algorithm 1]] with  '''REST''' = <math display="inline">{\mbox{PW}}_\mbox{x}</math>, <math>\nu=2</math> and bilinear prolongation was used. As the grid  becomes more refined, <math>\rho_{h} \rightarrow\rho_{loc} </math> for every orthotropy coefficient analyzed, what demonstrates the robustness of the methodology assessed.
 <div id='img-7'></div>
@@ Line 818: / Line 818: @@
-In order to assess the performance of the multigrid method with different anisotropy coefficients, a curve adjustment of the form <math>{{t}_{CPU}}=c{{N}^{p}}</math> [20] was made, where <math>p</math> represents the complexity order of the solver, ''N'' is the number of grid points and<math>N</math> <math>c</math> is a constant that depends on the method. The closer the value of <math>p</math> is to one, the better the performance of the method used. Ideally, multigrid presents <math>p=1</math>, what means that the CPU time grows linearly with the increase of ''N''. Results are shown in Table 2 for both restrictions assessed<math>N</math> (<math>\mbox{FW}</math> and  <math display="inline">{\mbox{PW}}_\mbox{x}</math>).
+In order to assess the performance of the multigrid method with different anisotropy coefficients, a curve adjustment of the form <math>{{t}_{CPU}}=c{{N}^{p}}</math> [20] was made, where <math>p</math> represents the complexity order of the solver, ''N'' is the number of grid points and<math>N</math> <math>c</math> is a constant that depends on the method. The closer the value of <math>p</math> is to one, the better the performance of the method used. Ideally, multigrid presents <math>p=1</math>, what means that the CPU time grows linearly with the increase of ''N''. Results are shown in [[#tab-2|Table 2]] for both restrictions assessed<math>N</math> (<math>\mbox{FW}</math> and  <math display="inline">{\mbox{PW}}_\mbox{x}</math>).
 One can observe in [[#tab-2|Table 2]] that, for every orthotropy employed, the multigrid method has a good performance, since <math>p\approx1</math> in every case. These results prove the efficiency and robustness of Algorithm 1, proposed in this work.

@@ Line 388: / Line 388: @@
 Double precision arithmetic was used for the simulations. The numerical codes were implemented in the Fortran 2003 language, using the Intel 9.1 Visual Fortran application. All numerical results were obtained in a computer with Intel Core i7 2.66 GHz processor, 16 GB RAM and Windows 8 operating system, 64-bit version.
-== 3. Local Fourier Analysis ==
+== 3. Local Fourier analysis ==
 LFA allows to predict the performance of the multigrid method as it supplies estimates of the convergence rate of its components. Therefore, LFA becomes a powerful tool in quantitative analysis and in the research of efficient multigrid methods.
@@ Line 464: / Line 464: @@
 |style="padding:10px;"| [[File:234.png|200px|centre|thumb]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;"| '''Figure 2'''. Low- (inner white area) and high-frequency (hatched area) areas.
+| colspan="1" style="padding:10px;"| '''Figure 2'''. Low- (inner white area) and high-frequency (hatched area) areas
 |}