Chavez et al 2020a - Revision history

Rimni at 13:36, 20 May 2021

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Rimni at 13:34, 20 May 2021

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Rimni: /* 1. The Motz problem */

2021-05-20T13:32:24Z

‎1. The Motz problem

Rimni at 13:43, 14 January 2021

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Dmota at 02:05, 14 January 2021

2021-01-14T02:05:29Z

Dmota at 02:02, 14 January 2021

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Dmota at 02:01, 14 January 2021

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Rimni at 13:47, 13 January 2021

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Rimni: /* References */

2021-01-13T13:32:13Z

‎References

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Rimni at 13:25, 13 January 2021

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@@ Line 340: / Line 340: @@
 |}
-For convenience, the ghost point <math display="inline">p_g</math> is calculated in such a way that <math display="inline">p_0</math> is the midpoint of <math display="inline">p_g</math> and <math display="inline">p_c</math>, the latter being the centroid of the nodes <math display="inline">p_0</math>, <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math>, and <math display="inline">p_5</math> (Figure [[#img-4|4]]).
+For convenience, the ghost point <math display="inline">p_g</math> is calculated in such a way that <math display="inline">p_0</math> is the midpoint of <math display="inline">p_g</math> and <math display="inline">p_c</math>, the latter being the centroid of the nodes <math display="inline">p_0</math>, <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math>, and <math display="inline">p_5</math> ([[#img-4|Figure 4]]).
 The value of <math display="inline">u(p_g)</math> given by the equation ([[#eq-13|13]]) is substituted in the approximation to the Laplacian (equation ([[#eq-7|7]]) at <math display="inline">p_0</math> using the same 7 points <math display="inline">p_g</math>, <math display="inline">p_0</math>, <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math>, and <math display="inline">p_5</math>, which eliminates the value of <math display="inline">u(p_g)</math>.

@@ Line 277: / Line 277: @@
-In the numerical tests, the interval <math display="inline">[1.5,2.5]</math> was considered for the exponent <math display="inline">\alpha </math>. For comparison purposes, the mapping was applied to the <math display="inline">x</math>-axis (Type I grids), to the <math display="inline">y</math>-axis (Type II grids), and both axes (Type III). An example of the grids corresponding to the different distributions is shown in Figure [[#img-2|2]].
+In the numerical tests, the interval <math display="inline">[1.5,2.5]</math> was considered for the exponent <math display="inline">\alpha </math>. For comparison purposes, the mapping was applied to the <math display="inline">x</math>-axis (Type I grids), to the <math display="inline">y</math>-axis (Type II grids), and both axes (Type III). An example of the grids corresponding to the different distributions is shown in [[#img-2| Figure 2]].
 <div id='img-2'></div>
@@ Line 290: / Line 290: @@
-The three kinds of grids define a regular location of neighbors at the inner nodes: star-shaped subgrids  with 6 neighbours that can be used to approximate the derivatives in the left-hand side of equation ([[#eq-6|6]]); an example of such subgrid can be seen in Figure [[#img-3|3]]. The Laplacian operator is approximated at the point <math display="inline">p_0</math>, and the corresponding neighbours are <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math>, <math display="inline">p_5</math>, and <math display="inline">p_6</math>. The values <math display="inline">h_1</math>, <math display="inline">h_2</math>, <math display="inline">k_1</math>, and <math display="inline">k_2</math> are the step size lengths in the stencil. Bearing in mind equation ([[#eq-5|5]]), this poses a problem with 6 equations and <math display="inline">k=7</math> unknowns; since the values <math display="inline">h_1</math>, <math display="inline">h_2</math>, <math display="inline">k_1</math>, and <math display="inline">k_2</math> are non zero, after preconditioning of equation ([[#eq-6|6]]) by scaling, the rank of <math display="inline">M</math> is equal to five,  which allows a robust calculation of the coefficients and leaves one degree of freedom available.
+The three kinds of grids define a regular location of neighbors at the inner nodes: star-shaped subgrids  with 6 neighbours that can be used to approximate the derivatives in the left-hand side of equation ([[#eq-6|6]]); an example of such subgrid can be seen in [[#img-3| Figure 3]]. The Laplacian operator is approximated at the point <math display="inline">p_0</math>, and the corresponding neighbours are <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math>, <math display="inline">p_5</math>, and <math display="inline">p_6</math>. The values <math display="inline">h_1</math>, <math display="inline">h_2</math>, <math display="inline">k_1</math>, and <math display="inline">k_2</math> are the step size lengths in the stencil. Bearing in mind equation ([[#eq-5|5]]), this poses a problem with 6 equations and <math display="inline">k=7</math> unknowns; since the values <math display="inline">h_1</math>, <math display="inline">h_2</math>, <math display="inline">k_1</math>, and <math display="inline">k_2</math> are non zero, after preconditioning of equation ([[#eq-6|6]]) by scaling, the rank of <math display="inline">M</math> is equal to five,  which allows a robust calculation of the coefficients and leaves one degree of freedom available.
 <div id='img-3'></div>
@@ Line 314: / Line 314: @@
 |}
-where <math display="inline">n=(l_x,l_y)</math> is the outer normal vector at <math display="inline">p_0</math>, is approximated using the scheme of equation ([[#eq-5|5]]) with <math display="inline">A=B=E=0</math> and <math display="inline">C=l_x</math>, <math display="inline">D=l_y</math> using 7 points: a ghost point <math display="inline">p_g</math>, and the nodes <math display="inline">p_0</math>, <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math> and <math display="inline">p_5</math> (Figure [[#img-4|4]]), which yields the equation ([[#eq-12|12]])
+where <math display="inline">n=(l_x,l_y)</math> is the outer normal vector at <math display="inline">p_0</math>, is approximated using the scheme of equation ([[#eq-5|5]]) with <math display="inline">A=B=E=0</math> and <math display="inline">C=l_x</math>, <math display="inline">D=l_y</math> using 7 points: a ghost point <math display="inline">p_g</math>, and the nodes <math display="inline">p_0</math>, <math display="inline">p_1</math>, <math display="inline">p_2</math>, <math display="inline">p_3</math>, <math display="inline">p_4</math> and <math display="inline">p_5</math> ([[#img-4|Figure 4]]), which yields the equation ([[#eq-12|12]])
 <span id="eq-12"></span>
@@ Line 357: / Line 357: @@
 ==4 Numerical tests==
-Several tests were performed  with <math display="inline">m</math> and <math display="inline">2m-1</math> nodes per side on the <math display="inline">y</math> and <math display="inline">x</math> directions, respectively, varying <math display="inline">m</math> from 31 to 351. Grids of the three types in the previous section were generated, and the Motz problem was solved using the proposed scheme for every grid size and distribution. An example of the finite difference discretization described in the previous section is sketched in Figure [[#img-5|5]].
+Several tests were performed  with <math display="inline">m</math> and <math display="inline">2m-1</math> nodes per side on the <math display="inline">y</math> and <math display="inline">x</math> directions, respectively, varying <math display="inline">m</math> from 31 to 351. Grids of the three types in the previous section were generated, and the Motz problem was solved using the proposed scheme for every grid size and distribution. An example of the finite difference discretization described in the previous section is sketched in [[#img-5|Figure 5]].
 <div id='img-5'></div>
@@ Line 382: / Line 382: @@
 was used, where <math display="inline">u_{i,j}</math> is the solution calculated using the difference scheme at the point <math display="inline">p_{i,j}</math>, <math display="inline">u</math> is the exact solution (equation [[#eq-1|(1)]]) at the same point, and <math display="inline">A_{i,j}</math> is the area of the rhombus defined by the nodes <math display="inline">p_{i+1,j}, p_{i-1,j}, p_{i,j+1}, p_{i,j-1}</math>.
-The measure of the roughness of the grid is given by the meh norm <math display="inline">h</math>; the log graph of the  mean square errors versus the grid norm for the type I and II grids are shown in Figures [[#img-6|6]] and [[#img-7|7]].  One can note that there is a clear convergence tendency on all the tests despite the boundary singularity. But, when only a nonuniform distribution either along the <math display="inline">x</math> or the <math display="inline">y</math>-axis is used,  the second-order approximation is lost, in a similar way as reported by Bernal using radial basis functions. The best results are obtained with the grids which are non-uniform on <math display="inline">x</math> and <math display="inline">y</math>, as can be seen in Figure [[#img-7|7]].
+The measure of the roughness of the grid is given by the meh norm <math display="inline">h</math>; the log graph of the  mean square errors versus the grid norm for the type I and II grids are shown in Figures [[#img-6|6]] and [[#img-7|7]].  One can note that there is a clear convergence tendency on all the tests despite the boundary singularity. But, when only a nonuniform distribution either along the <math display="inline">x</math> or the <math display="inline">y</math>-axis is used,  the second-order approximation is lost, in a similar way as reported by Bernal using radial basis functions. The best results are obtained with the grids which are non-uniform on <math display="inline">x</math> and <math display="inline">y</math>, as can be seen in  [[#img-7|Figure 7]].
 <div id='img-6'></div>

@@ Line 17: / Line 17: @@
 ==1. The Motz problem==
-The Motz problem is a benchmark for the Laplace equation problem that is often used for testing various special numerical methods; it was proposed in the literature for the solution of elliptic boundary value problems with boundary singularities. The boundary value problem is stated as follows (Figure [[#img-1|1]])
+The Motz problem is a benchmark for the Laplace equation problem that is often used for testing various special numerical methods; it was proposed in the literature for the solution of elliptic boundary value problems with boundary singularities. The boundary value problem is stated as follows ([[#img-1|Figure 1]])
 {| class="formulaSCP" style="width: 100%; text-align: left;"

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-[[#citeF-1|[1]]] Lu T. T., Hu H. Y., Li Z. C. Highly accurate solutions of Motz's and the cracked beam problems. Engineering Analysis with Boundary Elements, 28:1387–1403, 2004.
+[[#citeF-1|[1]]] Lu T.T., Hu H.Y., Li Z.C. Highly accurate solutions of Motz's and the cracked beam problems. Engineering Analysis with Boundary Elements, 28:1387–1403, 2004.
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+[[#citeF-2|[2]]] Li Z.C., Mathon R., Sermer P. Boundary methods for solving elliptic problems with singularities and interfaces. SIAM Journal of Numerical Analysis, 24:487–498, 1987.
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+[[#citeF-3|[3]]] Li Z.C. An approach for combining the Ritz-Galerkin and Finite-Element Methods. Journal of Approximation Theory, 39:132-152, 1983.
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+[[#citeF-4|[4]]] Arad M., Yosibash Z., Ben-Dor G.,  Yakhot A. Computing flux intensity factors by a boundary method for elliptic equations with singularities. Communications in Numerical Methods in Engineering, 14:657–670, 1998.
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+[[#citeF-5|[5]]] Georgiou G.C., Olson L., Smyrlis Y.S. A singular function boundary integral method for the Laplace equation. Communications in Numerical Methods in Engineering, 12:127-134, 1996.
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-[[#citeF-6|[6]]] Bernal F., Kindelan, M. Radial basis function solution of the Motz problem. Engineering Computations, 27: 606–620, 2010.
+[[#citeF-6|[6]]] Bernal F., Kindelan, M. Radial basis function solution of the Motz problem. Engineering Computations, 27:606–620, 2010.
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+[[#citeF-7|[7]]] Kansa E.J. Multiquadrics, a scattered data approximation scheme with applications to computational fluid dynamics II: solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers and Mathematics with Applications, 19:147–161, 1990.
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+[[#citeF-8|[8]]] Shu C., Ding H., Yeo K.S. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 192:941–954, 2003.
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-[[#citeF-9|[9]]] Hong W.T. Enriched Meshfree Method for an Accurate Numerical Solution of the Motz Problem. Advances in Mathematical Physics, 2016:1-12, 2016.
+[[#citeF-9|[9]]] Hong W.T. Enriched meshfree method for an accurate numerical solution of the Motz problem. Advances in Mathematical Physics, 2016:1-12, 2016.
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-[[#citeF-10|[10]]] Domínguez-Mota F. J., Mendoza-Armenta S., Tinoco-Guerrero G., Tinoco-Ruiz J. G. Finite difference schemes satisfying an optimality condition for the unsteady heat equation. Mathematics and Computers in Simulation, 106:76–83, 2014.
+[[#citeF-10|[10]]] Domínguez-Mota F.J., Mendoza-Armenta S., Tinoco-Guerrero G., Tinoco-Ruiz J.G. Finite difference schemes satisfying an optimality condition for the unsteady heat equation. Mathematics and Computers in Simulation, 106:76–83, 2014.
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+[[#citeF-12|[12]]] Tinoco-Guerrero G., Domínguez-Mota F.J., Gaona-Arias A., Ruiz-Zavala M.L., Tinoco-Ruiz J.G. A stability analysis for a generalized finite-difference scheme applied to the pure advection equation. Mathematics and Computers in Simulation, 147:293–300, 2018.
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+[[#citeF-13|[13]]] Benito J.J., Ureña F., Gavete L. Solving parabolic and hyperbolic equations by the generalized finite difference method. Journal of Computational and Applied Mathematics, 209:208–233, 2007.
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+[[#citeF-14|[14]]] Benito J.J., Ureña F., Gavete, L. Application of the generalized finite difference method to solve the advection-diffusion equation. Journal of Computational and Applied Mathematics, 235:1849–1855, 2011.
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-[[#citeF-15|[15]]] Benito J. J., Ureña F., Salete E., Gavete, L. A note on the application of the generalized finite difference method to seismic wave propagation in 2D. Journal of Computational and Applied Mathematics, 236:3016–3025, 2012.
+[[#citeF-15|[15]]] Benito J.J., Ureña F., Salete E., Gavete, L. A note on the application of the generalized finite difference method to seismic wave propagation in 2D. Journal of Computational and Applied Mathematics, 236:3016–3025, 2012.
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+[[#citeF-17|[17]]] Fan C.M., Li P.W. Generalized finite difference method for solving two-dimensional Burgers' equations. Procedia Engineering, 79:55–60, 2014.
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+[[#citeF-20|[20]]] Li P.W. , Chen W., Fu Z.J., Fan C.M. Generalized finite difference method for solving the double-diffusive natural convection in fluid-saturated porous media. Engineering Analysis with Boundary Elements 95:175–186, 2018.
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+[[#citeF-24|[24]]] Ivanenko, S. In Selected chapters on grid generation and applications. B. Azarenok (Ed.), Russian Academy of Sciences, 2009.
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+[[#citeF-27|[27]]] Domínguez-Mota F.J., Equihua M., Mendoza S., Tinoco-Ruiz J.G. Solución de ecuaciones diferenciales elípticas en regiones planas irregulares usando mallas convexas generadas por métodos variacionales empleando elementos finitos. Rev. Int. Mét. Num. Cálc. Dis. Ing., 26:187–194, 2010.

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 <div id="cite-24"></div>
-[[#citeF-24|[24]]] Ivanenko, S. In Selected chapters on grid generation and applications, B. Azarenok (Ed.), Russian academy of sciences, 2009.
+[[#citeF-24|[24]]] Ivanenko, S. In Selected chapters on grid generation and applications, B. Azarenok (Ed.), Russian Academy of Sciences, 2009.
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@@ Line 59: / Line 59: @@
 where <math display="inline">r</math> and <math display="inline">\theta </math> are the polar coordinates; its convergence ratio is 2, and therefore valid over the whole domain  (Figure [[#img-1|1]]).
-Several papers discuss the accurate numerical calculation of the coefficients <math display="inline">A_i</math>. Lu <span id='citeF-1'></span>[[#cite-1|[1]]] calculated them with 17 significant digits using a collocation Treftz method. Li ''et al.'' <span id='citeF-2'></span>[[#cite-2|[2]]] used a special  boundary approximation method to derive the leading coefficients of the asymptotic solution expansion. Similar approaches were followed by Li ''et al.''  <span id='citeF-3'></span>[[#cite-3|[3]]] and Arad ''et al.'' <span id='citeF-4'></span>[[#cite-4|[4]]], who  employed boundary methods combined with least-squares and penalty method techniques to handle the boundary information, and by Georgiou  <span id='citeF-5'></span>[[#cite-5|[5]]], who employed Lagrange multipliers.
+Several papers discuss the accurate numerical calculation of the coefficients <math display="inline">A_i</math>. Lu <span id='citeF-1'></span>[[#cite-1|[1]]] calculated them with 17 significant digits using a collocation Treftz method. Li ''et al.'' <span id='citeF-2'></span>[[#cite-2|[2]]] used a special  boundary approximation method to derive the leading coefficients of the asymptotic solution expansion. Similar approaches were followed by Li <span id='citeF-3'></span>[[#cite-3|[3]]] and Arad ''et al.'' <span id='citeF-4'></span>[[#cite-4|[4]]], who  employed boundary methods combined with least-squares and penalty method techniques to handle the boundary information, and by Georgiou  <span id='citeF-5'></span>[[#cite-5|[5]]], who employed Lagrange multipliers.
 Some other authors focus on solving the Motz problem using different numerical techniques. Bernal and Kindelan <span id='citeF-6'></span>[[#cite-6|[6]]] thoroughly discussed the performance of the radial basis function method in the solution of the Motz problem, including the global radial basis function method due to Kansa <span id='citeF-7'></span>[[#cite-7|[7]]], as well as  Shu's local  differential quadrature method based on radial basis functions <span id='citeF-8'></span>[[#cite-8|[8]]]. They showed that the exponential convergence obtained by increasing resolution and increasing the shape parameter is lost due to the presence of the singularity. The accuracy of the solution can be increased by using collocation at some boundary nodes to restore the convergence of the method, although it is necessary to use functions similar to the terms in equation ([[#eq-1|1]]), which are capable of capturing the behaviour of the solution near the discontinuity.