Tinoco Guerrero et al 2020a - Revision history

Scipediacontent at 06:58, 21 July 2022

2022-07-21T06:58:57Z

Scipediacontent at 06:51, 21 July 2022

2022-07-21T06:51:33Z

Zarate: Zarate moved page Review 838453312870 to Tinoco Guerrero et al 2020a

2020-03-09T08:13:24Z

Zarate moved page Review 838453312870 to Tinoco Guerrero et al 2020a

Gstinoco at 23:39, 6 March 2020

2020-03-06T23:39:59Z

Gstinoco at 23:17, 6 March 2020

2020-03-06T23:17:05Z

Gstinoco at 23:02, 6 March 2020

2020-03-06T23:02:41Z

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Gstinoco: Gstinoco moved page Draft Tinoco Guerrero 568708440 to Review 838453312870

2020-02-14T18:10:39Z

Gstinoco moved page Draft Tinoco Guerrero 568708440 to Review 838453312870

Gstinoco: Grammar mistakes and errata

2020-02-14T18:04:04Z

Grammar mistakes and errata

Gstinoco: Created page with "==A Generalized Finite Difference-Volume Hybrid Method Applied to Shallow-Water Equations== '''G. Tinoco-Guerrero*, F. J. Domínguez-Mota, J. G. Tino..."

2020-02-14T17:53:12Z

Created page with "==A Generalized Finite Difference-Volume Hybrid Method Applied to Shallow-Water Equations== '''G. Tinoco-Guerrero*, F. J. Domínguez-Mota, J. G. Tino..."

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@@ Line 1,034: / Line 1,034: @@
 Now, an adequate selection of <math display="inline">m</math>, the number of points, used by the scheme has to be done in order to represent different characteristics accurately. In this work, the selection of <math display="inline">m</math> has been done as follows: <math display="inline">m = 4</math> for <math display="inline">q</math>, and <math display="inline">m = 6</math> for <math display="inline">r</math> and <math display="inline">s</math>. The chosen stencils are shown in figure ([[#img-3|3]]).     <div id='img-3'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Review_838453312870-fig03.png|600px|'''Different stencils used by the scheme.''']]
+|[[Image:Tinoco_Guerrero_et_al_2020a_2538_Tinoco_Guerrero_et_al_2020a-fig03_a.png|600px|'''Different stencils used by the scheme.''']]
-|[[Image:Review_838453312870-fig05.png|292px|]]
+|[[Image:Tinoco_Guerrero_et_al_2020a_7111_Tinoco_Guerrero_et_al_2020a-fig03_b.png|292px|]]
 |- style="text-align: center; font-size: 75%;"
 | (3) '''Different stencils used by the scheme.'''
 |-
-| colspan="2"|[[Image:Review_838453312870-fig04.png|420px|]]
+| colspan="2"|[[Image:Tinoco_Guerrero_et_al_2020a_8525_Tinoco_Guerrero_et_al_2020a-fig03_c.png|420px|]]
 |}

@@ Line 61: / Line 61: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Review_838453312870-fig01.png|450px|'''Definition of bottom and free surface.''']]
+|[[Image:Tinoco_Guerrero_et_al_2020a_1041_Tinoco_Guerrero_et_al_2020a-fig01.png|450px|'''Definition of bottom and free surface.''']]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 1:''' '''Definition of bottom and free surface.'''

@@ Line 2: / Line 2: @@
 ==A Generalized Finite Difference-Volume Hybrid Method Applied to Shallow-Water Equations==
-'''G.&nbsp;Tinoco-Guerrero<sup>a,b</sup>, F.&nbsp;J.&nbsp;Domínguez-Mota<sup>a</sup>, J.&nbsp;G.&nbsp;Tinoco-Ruiz<sup>a</sup>, J.&nbsp;S.&nbsp;Lucas-Martínez<sup>a</sup>, N.&nbsp;S.&nbsp;Tinoco-Guerrero<sup>a</sup>, I.&nbsp;Mammarella<sup>b</sup>'''
+'''G.&nbsp;Tinoco-Guerrero<sup>a,b</sup>, F.&nbsp;J.&nbsp;Domínguez-Mota<sup>a</sup>, J.&nbsp;G.&nbsp;Tinoco-Ruiz<sup>a</sup>, J.&nbsp;S.&nbsp;Lucas-Martínez<sup>a</sup>, N.&nbsp;S.&nbsp;Tinoco-Guerrero<sup>a</sup>, M.&nbsp;Leppäranta<sup>b</sup>,I.&nbsp;Mammarella<sup>b</sup>'''
 -->

@@ Line 9: / Line 9: @@
 Due to the importance of the shallow-water equations in models of real-life phenomena, in recent years the study and model of problems that involve them have been the object of interest of many people. By reason of this, it is imperative to have efficient numerical methods to obtain an approximation of the solutions of the shallow-water equations.
-Several authors have worked in approximations using the well-known finite volume and finite element methods, nevertheless, even when these methods compute good approximations to real-life behavior, the computational cost is usually high, which could be limitation to the application of these methods.
+Several authors have worked in approximations using the well-known finite volume and finite element methods, nevertheless, even when these methods compute good approximations to real-life behavior, the computational cost is usually high, which could be a limitation to the application of these methods.
 This paper presents an explicit Generalized Finite Difference-Volume Hybrid approximation to the solution of the shallow-water equations, solved on irregular regions meshed with logically rectangular grids; the numerical results show the accuracy obtained with a low-cost implementation. The proposed scheme is a hybridization of a generalized finite difference scheme with the finite volume method.
@@ Line 789: / Line 789: @@
 |}
-To find the <math display="inline">\Gamma </math> values that satisfy these conditions requires to solve the systems
+To find the <math display="inline">\Gamma </math> values that satisfy these conditions requiress to solve the systems
 <span id="eq-23"></span>
@@ Line 826: / Line 826: @@
 |}
-it has to be noted that each system has <math display="inline">3</math> equations and <math display="inline">m+1</math> unknowns so, in general, in a suitable grid, there is a non trivial kernel. To select a solution, we use a subset of the normal equations of the corresponding least squares problem; then, considering the last <math display="inline">2</math> equations in each system
+it has to be noted that each system has <math display="inline">3</math> equations and <math display="inline">m+1</math> unknowns so, in general, in a suitable grid, there is a non trivial kernel. To select a solution, we use a subset of the normal equations of the corresponding least-squares problem; then, considering the last <math display="inline">2</math> equations in each system
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,315: / Line 1,315: @@
 <div id="cite-7"></div>
-'''[[#citeF-7|[7]]]''' Carlos Parés. (2006) "Numerical Nethods for Nonconservative Hyperbolic Systems: A Theoretical Framework", Volume 44. SIAM Journal on Numerical Analysis 1 300-321
+'''[[#citeF-7|[7]]]''' Carlos Parés. (2006) "Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework", Volume 44. SIAM Journal on Numerical Analysis 1 300-321
 <div id="cite-8"></div>

@@ Line 20: / Line 20: @@
 '''keywords'''
-hybrid methodfinite differencefinite volumeshallow-water equationsirregular regions    65M06, 65M22
+hybrid method; finite difference; finite volume; shallow-water equations; irregular regions
 ==1 Introduction==
@@ Line 70: / Line 70: @@
 |[[Image:Draft_Tinoco Guerrero_568708440-fig01.png|450px|'''Definition of bottom and free surface.''']]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1:''' '''Definition of bottom and free surface.'''
+| colspan="1" | '''Figure 1:''' '''Definition of the bottom and free surface.'''
 |}
@@ Line 191: / Line 191: @@
 ====2.1.1 Hybrid scheme for the conservative form====
-With this discretization of the region, the proposed scheme approximates the value of <math display="inline">q</math> at the center of each cell by using a classical finite difference approximation. For this case, the partial derivatives can be approximated at a central point <math display="inline">p_{(i, j)}</math> as
+With this discretization of the region, the proposed scheme approximates the value of <math display="inline">q</math>at the center of each cell by using a classical finite difference approximation. For this case, the partial derivatives can be approximated at a central point <math display="inline">p_{(i, j)}</math> as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 833: / Line 833: @@
 |}
-it has to be noted that each system has <math display="inline">3</math> equations and <math display="inline">m+1</math> unknowns so, in general, in a suitable grid, there is a non trivial kernel. To select a solution, we use a subset of the normal equations of the corresponding least squares problem; then, considering the last <math display="inline">2</math> equations in each system
+it has to be noted that each system has <math display="inline">3</math> equations and <math display="inline">m+1</math> unknowns so, in general, in a suitable grid, there is a non trivial kernel. To select a solution, we use a subset of the normal equations of the corresponding least-squares problem; then, considering the last <math display="inline">2</math> equations in each system
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,102: / Line 1,102: @@
 where <math display="inline">\min \Delta x</math> and <math display="inline">\min \Delta y</math> are the minimum values of <math display="inline">\Delta x</math> and <math display="inline">\Delta y</math> over the region and <math display="inline">c = 0.5\frac{t}{d}</math>. With this, the time interval <math display="inline">(0s,5s)</math> was divided into <math display="inline">50,000</math> steps.
-The total amounts of mass and momentum, over all the domain, are given at a time step <math display="inline">k</math> as
+The total amounts of mass and momentum, over  all the domain, are given at a time step <math display="inline">k</math> as
 {| class="formulaSCP" style="width: 100%; text-align: left;"

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