Balam et al 2023a - Revision history

Rimni at 13:33, 4 January 2024

2024-01-04T13:33:25Z

Rimni: /* 4. Two-dimensional problem */

2024-01-04T13:25:57Z

‎4. Two-dimensional problem

Rimni: /* References */

2024-01-04T13:20:26Z

‎References

Rimni: /* References */

2024-01-04T12:52:48Z

‎References

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Rimni at 11:21, 4 January 2024

2024-01-04T11:21:10Z

Rimni at 11:17, 4 January 2024

2024-01-04T11:17:11Z

Rimni at 11:16, 4 January 2024

2024-01-04T11:16:17Z

Rimni: /* 5.2.3 2D Poisson solution with large jump derivatives */

2024-01-04T11:15:09Z

‎5.2.3 2D Poisson solution with large jump derivatives

Rimni at 11:13, 4 January 2024

2024-01-04T11:13:56Z

Rimni at 11:09, 4 January 2024

2024-01-04T11:09:54Z

@@ Line 1,465: / Line 1,465: @@
+|
 | '''5.34'''
 |-style="text-align:center"
 !  <math display="inline">N</math>  !!  <math display="inline">L_\infty </math>-norm !!   Order !! <math display="inline">L_\infty </math>-norm !!   Order !! <math display="inline">L_\infty </math>-norm !!   Order
@@ Line 1,515: / Line 1,517: @@
+|
 | '''5.77'''
 |-style="text-align:center"
 !  <math display="inline">N</math>  !!  <math display="inline">L_\infty </math>-norm !!   Order !! <math display="inline">L_\infty </math>-norm !!   Order !! <math display="inline">L_\infty </math>-norm !!   Order

@@ Line 745: / Line 745: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\delta ^2_x u(x_i,y_j) = \dfrac{u_{i+1j}-2u_{ij}+u_{i-1j}}{h^2},</math>
+| style="text-align: center;" | <math>\begin{align} & \delta ^2_x u(x_i,y_j) = \dfrac{u_{i+1j}-2u_{ij}+u_{i-1j}}{h^2},\\
-|-
+& \delta ^4_x u(x_i,y_j) = \delta ^2_x\delta ^2_x u(x_i,y_j) = \dfrac{u_{i+2j}-4u_{i+1j}+6u_{ij}-4u_{i-1j}+u_{i-2j}}{h^4}\end{align}. </math>
-| style="text-align: center;" | <math> \delta ^4_x u(x_i,y_j) = \delta ^2_x\delta ^2_x u(x_i,y_j) = \dfrac{u_{i+2j}-4u_{i+1j}+6u_{ij}-4u_{i-1j}+u_{i-2j}}{h^4}. </math>
 |}
 |}

@@ Line 1,672: / Line 1,672: @@
 ==References==
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@@ Line 1,734: / Line 1,735: @@
 <div id="cite-21"></div>
-[[#citeF-21|[21]]]  Itzá Balam, R., Hernandez-Lopez, F., Trejo-Sánchez, J., & Uh Zapata, M. An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains. Mathematical Biosciences and Engineering: MBE 18(1) (2020), 22&#8211;56.
+[[#citeF-21|[21]]]  Itzá Balam R., Hernandez-Lopez F., Trejo-Sánchez J., Uh Zapata M. An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains. Mathematical Biosciences and Engineering: MBE 18(1):22&#8211;56, 2020.
 <div id="cite-22"></div>
-[[#citeF-22|[22]]]   Leveque R.J., & Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31(4) (1994), 1019-1044.
+[[#citeF-22|[22]]]   Leveque R.J.,  Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31(4):1019-1044, 1994.
 <div id="cite-23"></div>
-[[#citeF-23|[23]]]   Wiegmann A., &  Bube K.P. The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal. 37(3) (2000), 827&#8211;862.
+[[#citeF-23|[23]]]   Wiegmann A.,  Bube K.P. The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal., 37(3):827&#8211;862, 2000.
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-[[#citeF-24|[24]]]   Berthelsen, P. A. (2004). A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys. 197(1), 364&#8211;386.
+[[#citeF-24|[24]]]   Berthelsen P.A.  A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys., 197(1):364&#8211;386, 2004.
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-[[#citeF-25|[25]]]   Seo, J. H. & Mittal, R. A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries, J. Comput. Phys. 230 (2011), 1000&#8211;1019.
+[[#citeF-25|[25]]]   Seo J.H., Mittal R. A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries. J. Comput. Phys., 230:1000&#8211;1019, 2011.
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-[[#citeF-26|[26]]]    Ito, K., Li, Z., & Kyei, Y. Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains. SIAM Journal on Scientific Computing 27(1) (2005), 346-367.
+[[#citeF-26|[26]]]    Ito K., Li Z.,  Kyei Y. Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains. SIAM Journal on Scientific Computing, 27(1):346-367, 2005.
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-[[#citeF-27|[27]]]   Gibou, F., & Fedkiw, R. A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. 202(2) (2005), 577-601.
+[[#citeF-27|[27]]]   Gibou F., Fedkiw R. A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains with applications to the Stefan problem. J. Comput. Phys., 202(2):577-601, 2005.
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-[[#citeF-28|[28]]]   Linnick, M. N., & Fasel, H. F. A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. Comput. Phys. 204(1) (2005), 157-192.
+[[#citeF-28|[28]]]   Linnick M.N.,  Fasel H.F. A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. Comput. Phys., 204(1):157-192, 2005.
 <div id="cite-29"></div>
-[[#citeF-29|[29]]]    Zhou, Y. C., Zhao, S., Feig, M., & Wei, G. W. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213(1) (2006), 1&#8211;30.
+[[#citeF-29|[29]]]    Zhou Y.C., Zhao S., Feig M., Wei G.W. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys., 213(1):1&#8211;30, 2006.
 <div id="cite-30"></div>
-[[#citeF-30|[30]]]   Zhong, X. A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity. J. Comput. Phys. 225(1) (2007), 1066-1099.
+[[#citeF-30|[30]]]   Zhong X. A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity. J. Comput. Phys., 225(1):1066-1099, 2007.
 <div id="cite-31"></div>
-[[#citeF-31|[31]]]   Feng, X., Li, Z., & Qiao, Z. High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients. J. of Comput. Math. (2011) 324&#8211;340.
+[[#citeF-31|[31]]]   Feng X., Li Z., Qiao Z. High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients. J. of Comput. Math., 29(3):324&#8211;340, 2011.
 <div id="cite-32"></div>
-[[#citeF-32|[32]]]    Pan, K., He, D., & Li, Z. A high order compact FD framework for elliptic BVPs involving singular sources, interfaces, and irregular domains. J. Sci. Comput. 88(3) (2021), 1&#8211;25.
+[[#citeF-32|[32]]]    Pan K., He D., Li Z. A high order compact FD framework for elliptic BVPs involving singular sources, interfaces, and irregular domains. J. Sci. Comput., 88(3):1&#8211;25, 2021.
 <div id="cite-33"></div>
-[[#citeF-33|[33]]]    Colnago, M., Casaca, W., & de Souza, L. F. A high-order immersed interface method free of derivative jump conditions for Poisson equations on irregular domains. J. Comput. Phys. 423 (2020), 109791.
+[[#citeF-33|[33]]]    Colnago M., Casaca W., de Souza L.F. A high-order immersed interface method free of derivative jump conditions for Poisson equations on irregular domains. J. Comput. Phys., 423:109791, 2020.
 <div id="cite-34"></div>
-[[#citeF-34|[34]]]    Feng, Q., Han, B., & Minev, P. Sixth order compact finite difference schemes for Poisson interface problems with singular sources. Computers & Mathematics with Applications 99 (2021), 2-25.
+[[#citeF-34|[34]]]    Feng Q., Han B., Minev P. Sixth order compact finite difference schemes for Poisson interface problems with singular sources. Computers & Mathematics with Applications, 99:2-25, 2021.
 <div id="cite-35"></div>
-[[#citeF-35|[35]]]           Claerbout, J.F. The craft of wave-field extrapolation, in Imaging the Earth's Interior, Blackwell Scientific Publications, Oxford (1985), 260-265.
+[[#citeF-35|[35]]]       Claerbout J.F. The craft of wave-field extrapolation. In Imaging the Earth's Interior, Blackwell Scientific Publications, Oxford, pp. 260-265, 1985.
 <div id="cite-36"></div>
-[[#citeF-36|[36]]]    Liu, Y., & Sen, M. K. A practical implicit finite-difference method: examples from seismic modeling. Journal of Geophysics and Engineering 6(3) (2009),  231.
+[[#citeF-36|[36]]]    Liu Y., Sen M.K. A practical implicit finite-difference method: examples from seismic modeling. Journal of Geophysics and Engineering, 6(3):231-249, 2009.
 <div id="cite-37"></div>
-[[#citeF-37|[37]]]  W.F. Spotz, High-order compact finite difference schemes for computational mechanics Doctoral dissertation, The University of Texas at Austin, 1995.
+[[#citeF-37|[37]]]   Spotz W.F. High-order compact finite difference schemes for computational mechanics. Ph.D. Thesis,  University of Texas at Austin, 1995.
 <div id="cite-38"></div>
-[[#citeF-38|[38]]]   Xu S., & Wang Z.J. Systematic Derivation of Jump Conditions for the Immersed Interface Method in Three-Dimensional Flow Simulation, J. Sci. Comput. 27(6) (2006), 1948-1980.
+[[#citeF-38|[38]]]   Xu S., Wang Z.J. Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation. J. Sci. Comput., 27(6):1948-1980, 2006.
 <div id="cite-39"></div>
-[[#citeF-39|[39]]]  Feng, X., & Li, Z. Simplified immersed interface methods for elliptic interface problems with straight interfaces. Num. Meth. for Par. Diff. Eqs. 28(1) (2012), 188&#8211;203.
+[[#citeF-39|[39]]]  Feng X., Li Z. Simplified immersed interface methods for elliptic interface problems with straight interfaces. Num. Meth. for Par. Diff. Eqs., 28(1):188&#8211;203, 2012.
 <div id="cite-40"></div>
-[[#citeF-40|[40]]]  Li, Z. Immersed interface methods for moving interface problems. Numerical algorithms 14(4) (1997), 269-293.
+[[#citeF-40|[40]]]  Li Z. Immersed interface methods for moving interface problems. Numerical algorithms, 14(4):269-293, 1997.
 <div id="cite-41"></div>
-[[#citeF-41|[41]]]  Huang, H., & Li, Z. (1999). Convergence analysis of the immersed interface method. IMA Journal of Numerical Analysis, 19(4), 583-608.
+[[#citeF-41|[41]]]  Huang H., Li Z.  Convergence analysis of the immersed interface method. IMA Journal of Numerical Analysis, 19(4):583-608, 1999.

@@ Line 307: / Line 307: @@
 |}
-where <math display="inline">C_I</math> is defined as equation [[#eq-11|(11)]]. Substituting [[#eq-15|(15)]] into a standard central scheme for the second-order derivative, it yields
+where <math display="inline">C_I</math> is defined as Eq. [[#eq-11|(11)]]. Substituting [[#eq-15|(15)]] into a standard central scheme for the second-order derivative, it yields
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 1,667: / Line 1,667: @@
 |}
-==6 Conclusions==
+==6. Conclusions==
 The present paper introduces a new sixth-order immersed interface combined with an implicit finite difference to solve 2D Poisson problems with straight interfaces. The resulting numerical method is <math display="inline">O(h^6)</math> at regular points, and <math display="inline">O(h^5)</math> at irregular points. Furthermore, a fourth-order immersed interface method is obtained as a particular case of the proposed scheme. This paper also presents a numerical technique to handle the boundaries in the Poisson problem. The global accuracy of the sixth-order was demonstrated using several numerical examples. As expected, this approach does not depend on the interface position. For future work, the proposed approximation will be used to solve more general elliptic equations and interface shapes, and time-dependent problems in higher dimensions.

@@ Line 1,594: / Line 1,594: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\textrm{Example 2.E:}  u(x,y) = \begin{cases}0 & x\leq x_\alpha ,\\ 2+\ln (x(1+y^2)) & x_\alpha{<}x,\\ \end{cases} \quad  \textrm{where } \quad   x_\alpha = 0.5.   </math>
+| style="text-align: center;" | <math>\textrm{Example 2.E:}\quad  u(x,y) = \begin{cases}0 & x\leq x_\alpha ,\\ 2+\ln (x(1+y^2)) & x_\alpha{<}x,\\ \end{cases} \quad  \textrm{where } \quad   x_\alpha = 0.5.   </math>
 |}
 |}

@@ Line 1,656: / Line 1,656: @@
+|
 | '''5.21'''
 |}
 <div id='img-11'></div>
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:75%;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Balam_338770048-Fig_Exa4_2D_ABC_solcloud4th6thB.png|600px]]
+|style="text-align: center;padding:10px;"| [[Image:Draft_Balam_338770048-Fig_Exa4_2D_ABC_solcloud4th6thB.png|700px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 11'''. (a) Numerical and exact solution of Example 2.E using <math>N = 80</math>. (b)-(c) Convergence error analysis of IFD-IIM and HIFD-IIM, respectively, using different grid resolutions from <math>N=10</math> to <math>100</math>

@@ Line 1,594: / Line 1,594: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\textrm{Example 2.E:}  u(x,y) = \begin{cases}0 & x\leq x_\alpha ,\\ 2+\ln (x(1+y^2)) & x_\alpha{<}x,\\ \end{cases} \quad  \textrm{where}   x_\alpha = 0.5.   </math>
+| style="text-align: center;" | <math>\textrm{Example 2.E:}  u(x,y) = \begin{cases}0 & x\leq x_\alpha ,\\ 2+\ln (x(1+y^2)) & x_\alpha{<}x,\\ \end{cases} \quad  \textrm{where } \quad   x_\alpha = 0.5.   </math>
 |}
 |}
@@ Line 1,600: / Line 1,600: @@
 Here, the jump derivatives for all points at the interface are <math display="inline">[u_x] = 2</math>, <math display="inline">[u_{xx}] = -4</math>, <math display="inline">[u_{xxx}] =16</math>, <math display="inline">[u_{xxxx}] = -96</math>, <math display="inline">[u_{xxxxx}] =  768</math>, and <math display="inline">[u_{xxxxxx}] = -7680</math>. Is important to remark that opposite to the previous examples, the jump derivatives increase rapidly. This behavior makes the problem challenging to solve.
-Table [[#table-5|5]] and Fig.[[#img-11|11]] show the convergence analysis for Example 2.E. Numerical results show that Example 2.E has more variability in error than previous examples. However, the order of each technique is close to the proposed one. These findings also confirm that local truncation error depends not only on <math display="inline">h_R</math> and <math display="inline">h_L</math> but also on the jump magnitudes.
+[[#table-5|Table 5]] and [[#img-11|Figure 11]] show the convergence analysis for Example 2.E. Numerical results show that Example 2.E has more variability in error than previous examples. However, the order of each technique is close to the proposed one. These findings also confirm that local truncation error depends not only on <math display="inline">h_R</math> and <math display="inline">h_L</math> but also on the jump magnitudes.
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+<span id='table-5'></span>
-|+ style="font-size: 75%;" |<span id='table-5'></span>'''Table. 5''' Convergence analysis of Example 2.E.
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
-|-
+|-style="text-align:center"
-| style="text-align: right;" |
+!  <math display="inline">N</math>  !!  <math display="inline">L_\infty </math>-norm !!   Order !! <math display="inline">L_\infty </math>-norm !!   Order !! <math display="inline">L_\infty </math>-norm !!   Order
+|-style="text-align:center"
-−
-|-
-| <math display="inline">N</math>
-| <math display="inline">L_\infty </math>-norm
-|  Order
-| <math display="inline">L_\infty </math>-norm
-|  Order
-| <math display="inline">L_\infty </math>-norm
-|  Order
 |-
 | 10
 |  1.11e-02
@@ Line 1,625: / Line 1,616: @@
 |  1.38e-04
 |  &#8211;-
-|-
+|-style="text-align:center"
 | 20
 |  1.80e-03
@@ Line 1,633: / Line 1,624: @@
 |  2.48e-06
 |  5.80
-|-
+|-style="text-align:center"
 | 40
 |  5.64e-04
@@ Line 1,641: / Line 1,632: @@
 |  9.43e-08
 |  4.72
-|-
+|-style="text-align:center"
 | 80
 |  1.57e-04
@@ Line 1,649: / Line 1,640: @@
 |  2.61e-09
 |  5.17
-|-
+|-style="text-align:center"
 | 160
 |  4.11e-05
@@ Line 1,657: / Line 1,648: @@
 |  6.10e-11
 |  5.42
-|-
+|-style="text-align:center"
 | LSM
+|
@@ Line 1,668: / Line 1,659: @@
+|
 |}
 <div id='img-11'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-
-|[[Image:Draft_Balam_338770048-Fig_Exa4_2D_ABC_solcloud4th6thB.png|588px|Numerical and exact solution of Example 2.E using N = 80; (b),(c) convergence error analysis of IFD-IIM and HIFD-IIM respectively, using different grid resolutions from N=10 to 100.]]
+| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 11'''. (a) Numerical and exact solution of Example 2.E using <math>N = 80</math>. (b)-(c) Convergence error analysis of IFD-IIM and HIFD-IIM, respectively, using different grid resolutions from <math>N=10</math> to <math>100</math>
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 11:''' Numerical and exact solution of Example 2.E using <math>N = 80</math>; (b),(c) convergence error analysis of IFD-IIM and HIFD-IIM respectively, using different grid resolutions from <math>N=10</math> to <math>100</math>.
 |}

@@ Line 1,389: / Line 1,389: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\textrm{Example 2.B:}  \quad u(x,y) = \begin{cases}\sin (x)e^y & x\leq x_\alpha ,\\ -x^2e^y + 2& x_\alpha{<}x,\\ \end{cases}    \qquad \quad \quad \quad \textrm{where }    x_\alpha = 0.  </math>
+| style="text-align: center;" | <math>\textrm{Example 2.B:}  \quad u(x,y) = \begin{cases}\sin (x)e^y & x\leq x_\alpha ,\\ -x^2e^y + 2& x_\alpha{<}x,\\ \end{cases}    \qquad \quad \quad \quad \textrm{where }  \quad   x_\alpha = 0.  </math>
 |-
 | style="text-align: center;" | <math> \textrm{Example 2.C:}  \quad u(x,y) = \begin{cases}\sin (x)\sin (y) & x\leq x_\alpha ,\\ e^x\sin (3y) + 1 & x_\alpha{<}x,\\ \end{cases}   \quad \quad \quad  \textrm{where } \quad   x_\alpha = 0.4.  </math>