Puc et al 2023a - Revision history

Gstinoco at 00:37, 2 December 2023

2023-12-02T00:37:29Z

Muhzapata at 13:26, 18 November 2023

2023-11-18T13:26:48Z

Gstinoco: Gstinoco moved page Review 499745022132 to Puc et al 2023a

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Gstinoco moved page Review 499745022132 to Puc et al 2023a

Muhzapata at 20:11, 6 November 2023

2023-11-06T20:11:07Z

Muhzapata at 20:07, 6 November 2023

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Muhzapata at 20:02, 6 November 2023

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Muhzapata at 19:54, 6 November 2023

2023-11-06T19:54:59Z

Muhzapata at 19:49, 6 November 2023

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Muhzapata at 19:44, 6 November 2023

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Muhzapata at 19:37, 6 November 2023

2023-11-06T19:37:19Z

@@ Line 4: / Line 4: @@
 '''Heidy Escamilla Puc<sup>a</sup>, Reymundo Itzá Balam<sup>b,c</sup>, Miguel Uh Zapataangeluh@cimat.mx<sup>b,c</sup>'''
 -->
 ==1 Introduction==

@@ Line 442: / Line 442: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>C_i =  \begin{cases}- \dfrac{1}{h^2}\left([u]+h_R\left[u_{x}\right]+\dfrac{h_R^2}{2} + b \left(2 h_R^3\left[u_{xxx}\right]+\dfrac{h_R^4}{2}\left[u_{xxxx}\right]\right)\right), & i=I,\\[4mm]   \dfrac{1}{h^2} \left([u]+h_L\left[u_{x}\right]+\dfrac{h_L^2}{2} + b \left(2h_L^3\left[u_{xxx}\right]+\dfrac{h_L^4}{2}\left[u_{xxxx}\right]\right)\right), & i=I+1,\\[2mm]    0, & \textrm{else where,}  \end{cases}  </math>
+| style="text-align: center;" | <math>C_i =  \begin{cases}- \dfrac{1}{h^2}\left([u]+h_R\left[u_{x}\right]+\dfrac{h_R^2}{2} \left[u_{xx}\right] + b \left(2 h_R^3\left[u_{xxx}\right]+\dfrac{h_R^4}{2}\left[u_{xxxx}\right]\right)\right), & i=I,\\[4mm]   \dfrac{1}{h^2} \left([u]+h_L\left[u_{x}\right]+\dfrac{h_L^2}{2}\left[u_{xx}\right] + b \left(2h_L^3\left[u_{xxx}\right]+\dfrac{h_L^4}{2}\left[u_{xxxx}\right]\right)\right), & i=I+1,\\[2mm]    0, & \textrm{else where,}  \end{cases}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (23)
@@ Line 484: / Line 484: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathfrak{C}_i =  \begin{cases}  \dfrac{1}{h^2}\left([u]+h_R\left[u_{x}\right]+\dfrac{h_R^2}{2} \right)& \\ \qquad  - b \left\{[f] + h_R\left(- \dfrac{2h_R^2}{h^2}+1\right)\left[f_x\right]+\dfrac{h_R^2}{2} \left(-\dfrac{h_R^2}{h^2}+1\right)\left[f_{xx}\right]\right\}, & i=I, \\[2mm] - \dfrac{1}{h^2} \left([u]+h_L\left[u_{x}\right]+\dfrac{h_L^2}{2} \right)& \\ \qquad  + b \left\{[f] + h_L\left(- \dfrac{2h_L^2}{h^2} +1\right)\left[f_x\right]+ \dfrac{h_L^2}{2} \left(-\dfrac{h_L^2}{h^2}+1\right)\left[f_{xx}\right]\right\}, & i=I+1,\\[4mm]    0, & \textrm{else where}. \end{cases}  </math>
+| style="text-align: center;" | <math>\mathfrak{C}_i =  \begin{cases}  \dfrac{1}{h^2}\left([u]+h_R\left[u_{x}\right]+\dfrac{h_R^2}{2}\left[f\right] \right)& \\ \qquad  - b \left\{[f] + h_R\left(- \dfrac{2h_R^2}{h^2}+1\right)\left[f_x\right]+\dfrac{h_R^2}{2} \left(-\dfrac{h_R^2}{h^2}+1\right)\left[f_{xx}\right]\right\}, & i=I, \\[2mm] - \dfrac{1}{h^2} \left([u]+h_L\left[u_{x}\right]+\dfrac{h_L^2}{2}\left[f\right] \right)& \\ \qquad  + b \left\{[f] + h_L\left(- \dfrac{2h_L^2}{h^2} +1\right)\left[f_x\right]+ \dfrac{h_L^2}{2} \left(-\dfrac{h_L^2}{h^2}+1\right)\left[f_{xx}\right]\right\}, & i=I+1,\\[4mm]    0, & \textrm{else where}. \end{cases}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (26)

@@ Line 219: / Line 219: @@
 Finally, we get [[#eq-6|(6)]] which complete the proof. The same procedure can be applied for the proof at <math display="inline">x_{I+1}</math>. We refer to the reader to the work of Itza Balam and Uh Zapata <span id='citeF-14'></span>[[#cite-14|[14]]] for more details. ♦
-<span id='theorem-EFD'></span>Remark 1: If <math display="inline">b=0</math>, then Theorem [[#theorem-teo:regular|1]] yields the standard second-order finite-difference method for regular points given by
+<span id='theorem-EFD'></span>'''Remark 1:''' If <math display="inline">b=0</math>, then Theorem [[#theorem-teo:regular|1]] yields the standard second-order finite-difference method for regular points given by
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 491: / Line 491: @@
 Thus contribution <math display="inline">\mathfrak{C}_i</math> depends only on the principal jump conditions, <math display="inline">[u]</math> and <math display="inline">[u_x]</math>, and right-hand side jumps: <math display="inline">[f]</math>, <math display="inline">\left[f_{x}\right]</math>, and <math display="inline">\left[f_{xx} \right]</math>. The additional jumps derivatives from the right-hand side can be approximated using the current values of <math display="inline">f</math>. In this paper, we will assume that we know them.
-Remark 2: For the 1D Poisson problem, a global second-order IIM (<math display="inline">b=0</math>) only requires to know the principal jump conditions <math display="inline">[u]</math>, <math display="inline">\left[u_{x}\right]</math> and <math display="inline">\left[f\right]</math>.
+'''Remark 2:''' For the 1D Poisson problem, a global second-order IIM (<math display="inline">b=0</math>) only requires to know the principal jump conditions <math display="inline">[u]</math>, <math display="inline">\left[u_{x}\right]</math> and <math display="inline">\left[f\right]</math>.
-Remark 3: If <math display="inline">h_R/h=1</math>, then <math display="inline">h_L=0</math> and both weight terms next to second-order derivative jump of <math display="inline">f</math> are equal to zero in [[#eq-26|(26)]]. Thus, we do not require to know jump condition <math display="inline">\left[f_{xx}\right]</math> to obtain a fourth-order method when the interface is located at a grid point.
+'''Remark 3:''' If <math display="inline">h_R/h=1</math>, then <math display="inline">h_L=0</math> and both weight terms next to second-order derivative jump of <math display="inline">f</math> are equal to zero in [[#eq-26|(26)]]. Thus, we do not require to know jump condition <math display="inline">\left[f_{xx}\right]</math> to obtain a fourth-order method when the interface is located at a grid point.
 ==4 Heat equation with a fixed interface==
@@ Line 643: / Line 643: @@
 where <math display="inline">r=\frac{1}{2}\Delta t/h^2</math> and <math display="inline">\mathfrak{C}_i</math> is given by [[#eq-36|(36)]]. Here, we have that <math display="inline">\mathfrak{C}_i=0</math>  for regular points.
-Remark 4: The IFD-IIM [[#eq-37|(37)]] is unconditionally stable and the local truncation error is of <math display="inline">O(\Delta t^2 + h^4)</math> and <math display="inline">O(\Delta t^2 + h^3)</math> for regular and irregular grid points, respectively. Thus a global fourth-order method is expected by taking <math display="inline">\Delta t = O(h^2)</math>.
+'''Remark 4:''' The IFD-IIM [[#eq-37|(37)]] is unconditionally stable and the local truncation error is of <math display="inline">O(\Delta t^2 + h^4)</math> and <math display="inline">O(\Delta t^2 + h^3)</math> for regular and irregular grid points, respectively. Thus a global fourth-order method is expected by taking <math display="inline">\Delta t = O(h^2)</math>.
-Remark 5: As the Crank-Nicolson method is implicit in time, the IFD-IIM solves a linear system at each time step. To address this efficiently, we employed Thomas' algorithm, given that the resulting matrix is tridiagonal. Furthermore, the implicit method in space preserves the original structure of this system of equations, thus yielding a higher-order method without compromising the efficiency of the standard scheme.
+'''Remark 5:''' As the Crank-Nicolson method is implicit in time, the IFD-IIM solves a linear system at each time step. To address this efficiently, we employed Thomas' algorithm, given that the resulting matrix is tridiagonal. Furthermore, the implicit method in space preserves the original structure of this system of equations, thus yielding a higher-order method without compromising the efficiency of the standard scheme.
-Remark 6: In addition to accounting for the contributions of the source term <math display="inline">f</math> and its derivatives, the finite-difference scheme presented in equation [[#eq-37|(37)]] requires additional knowledge of the jump conditions for the solution given by <math display="inline">\left[u_{xx}\right]</math>, <math display="inline">\left[u_{xxx}\right]</math>, and <math display="inline">\left[u_{xxxx}\right]</math>. It is worth noting that, although not presented here, there are techniques available for deriving all the necessary jump conditions from the principal jump conditions <span id='citeF-14'></span>[[#cite-14|[14]]].
+'''Remark 6:''' In addition to accounting for the contributions of the source term <math display="inline">f</math> and its derivatives, the finite-difference scheme presented in equation [[#eq-37|(37)]] requires additional knowledge of the jump conditions for the solution given by <math display="inline">\left[u_{xx}\right]</math>, <math display="inline">\left[u_{xxx}\right]</math>, and <math display="inline">\left[u_{xxxx}\right]</math>. It is worth noting that, although not presented here, there are techniques available for deriving all the necessary jump conditions from the principal jump conditions <span id='citeF-14'></span>[[#cite-14|[14]]].
 ==5 Numerical results for the Poisson equation==

@@ Line 70: / Line 70: @@
 The following two Theorems state the main results to approximate the second-order derivative using high-order schemes for regular and irregular points.
-<span id='theorem-teo:regular'></span>Theorem 1:  '''Regular points''' <span id='citeF-34'></span><span id='citeF-35'></span>[[#cite-34|[34,35]]]. Let us consider a real-valued function <math display="inline">u</math> with an interface <math display="inline">x_\alpha </math> such that <math display="inline">x_I\leq x_\alpha{<}x_{I+1}</math>. Then <math display="inline">u_{xx}</math> can be approximated by a fourth-order IFD scheme
+<span id='theorem-teo:regular'></span> '''Theorem 1: Regular points''' <span id='citeF-34'></span><span id='citeF-35'></span>[[#cite-34|[34,35]]]. Let us consider a real-valued function <math display="inline">u</math> with an interface <math display="inline">x_\alpha </math> such that <math display="inline">x_I\leq x_\alpha{<}x_{I+1}</math>. Then <math display="inline">u_{xx}</math> can be approximated by a fourth-order IFD scheme
 <span id="eq-2"></span>
@@ Line 138: / Line 138: @@
 It is important to remark that formula [[#eq-2|(2)]] is applicable exclusively for regular points. In order to address this limitation for the two irregular points near the interface, we introduce a modified IFD scheme using the IIM, specifically tailored to handle discontinuous solutions. Furthermore, instead of having a fourth-order local truncation error for the irregular points, we proceed as other IIMs <span id='citeF-22'></span><span id='citeF-23'></span><span id='citeF-24'></span><span id='citeF-29'></span><span id='citeF-33'></span><span id='citeF-32'></span>[[#cite-22|[22,23,24,29,33,32]]] by taking one order lower at these points. We will numerically show that the global order of convergence can be still <math display="inline">O(h^4)</math> even if the local truncation error at <math display="inline">i=I</math> and <math display="inline">i=I+1</math> is <math display="inline">O(h^{3})</math>.
-<span id='theorem-teo:irregular'></span>Theorem 2:  '''Irregular points''' <span id='citeF-14'></span>[[#cite-14|[14]]]. Let us consider the known jump conditions
+<span id='theorem-teo:irregular'></span> '''Theorem 2: Irregular points''' <span id='citeF-14'></span>[[#cite-14|[14]]]. Let us consider the known jump conditions
 <span id="eq-5"></span>

@@ Line 9: / Line 9: @@
 This work forms the foundation for addressing high-order immersed interface methods to solve interface problems and enables us to conduct in-depth examination of this theory. Here, we focus on the introduction a fourth-order finite-difference formulation to approximate the second-order derivative of discontinuous functions. The approach is based on the combination of a high-order implicit formulation and the immersed interface method. The idea is to modify the standard schemes by introducing additional contribution terms based on jump conditions. These contributions are calculated only at grid points where the stencil intersects with the interface. Here, we discuss the issues of implementing the one-dimensional Poisson equation and the heat conduction equation with discontinuous solutions as a three-point stencil for each grid point on the computational domain. In both cases, the resulting discretization approach yields a tridiagonal linear system with matrix coefficients identical to those employed for smooth solutions. We present several numerical experiments to verify the feasibility and accuracy of the method. Thus, this high-order method provides an attractive numerical framework that can efficiently lead to the solution to more complex problems.
-''Keywords:'' Immersed interface method; Implicit finite difference; Fourth-order accuracy; Poisson equation; Heat conduction equation.
+''Keywords:'' Immersed interface method; Implicit finite difference; Fourth-order accuracy; Poisson equation; Heat conduction equation
 ==2 Introduction==

← Older revision	Revision as of 02:55, 16 November 2023
(No difference)

@@ Line 1,171: / Line 1,171: @@
 |+ style="font-size: 75%;" |<span id='table-4'></span>'''Table. 4''' Convergence analysis of Example 5 using the IFD-IIM for the heat equation with a discontinuous solution.
+!
-! colspan="4" |<math display="inline">x_α=0.4</math>
+! colspan="4" |<math display="inline">x_\alpha=0.4</math>
-! colspan="4" |<math display="inline">x_α=0.63</math>
+! colspan="4" |<math display="inline">x_\alpha=0.63</math>
 |-
 ! <math display="inline">N</math>
@@ Line 1,370: / Line 1,370: @@
 |+ style="font-size: 75%;" |<span id='table-6'></span>'''Table. 6''' Convergence analysis of Example 6 using the IFD-IIM for the heat equation with a general discontinuous solution.
+!
-! colspan="4" |<math display="inline">x_α=0.4</math>
+! colspan="4" |<math display="inline">x_\alpha=0.4</math>
-! colspan="4" |<math display="inline">x_α=0.63</math>
+! colspan="4" |<math display="inline">x_\alpha=0.63</math>
 |-
 ! <math display="inline">N</math>

@@ Line 874: / Line 874: @@
 ! <math display="inline">L_2</math>-norm
 !  Order
 |-
 | 10
@@ Line 1,178: / Line 1,168: @@
 |}
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:99%;"
 |+ style="font-size: 75%;" |<span id='table-4'></span>'''Table. 4''' Convergence analysis of Example 5 using the IFD-IIM for the heat equation with a discontinuous solution.
 |-
-|     0.99 @|c|cc|cc|cc|cc|
+! <math display="inline">N</math>
-| style="text-align: right;" |
+! <math display="inline">L_\infty </math>-norm
+!  Order
-|-
+! <math display="inline">L_2</math>-norm
-| <math display="inline">N</math>
+!  Order
-| <math display="inline">L_\infty </math>-norm
+! <math display="inline">L_\infty </math>-norm
-|  Order
+!  Order
-| <math display="inline">L_2</math>-norm
+! <math display="inline">L_2</math>-norm
-|  Order
+!  Order
-| <math display="inline">L_\infty </math>-norm
-|  Order
-| <math display="inline">L_2</math>-norm
-|  Order
 |-
 | 10
@@ Line 1,255: / Line 1,244: @@
 |  '''3.96'''
 |-
 |}
 From Table&nbsp;[[#table-5|5]], we can find the results when the time step is chosen as <math display="inline">\Delta t = h</math>. As expected, the proposed IFD-IIM is stable and has two-order convergence either we take a second- or fourth-order approximation in space. We remark that this is a property of the selected time integration method. For instance, similar findings as Table&nbsp;[[#table-5|5]] can be obtained for smooth solutions.
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:99%;"
 |+ style="font-size: 75%;" |<span id='table-5'></span>'''Table. 5''' Convergence analysis of Example 5 testing the heat equation with <math>\Delta t=h</math>.
 |-
-|      0.99 @|c|cc|cc|cc|cc|
+! <math display="inline">N</math>
+! <math display="inline">L_\infty </math>-norm
+!  Order
-|-
+! <math display="inline">L_2</math>-norm
-| <math display="inline">N</math>
+!  Order
-| <math display="inline">L_\infty </math>-norm
+! <math display="inline">L_\infty </math>-norm
-|  Order
+!  Order
-| <math display="inline">L_2</math>-norm
+! <math display="inline">L_2</math>-norm
-|  Order
+!  Order
-| <math display="inline">L_\infty </math>-norm
-|  Order
-| <math display="inline">L_2</math>-norm
-|  Order
 |-
 | 10
@@ Line 1,337: / Line 1,324: @@
 |  '''1.95'''
 |-
 |}
@@ Line 1,381: / Line 1,367: @@
 |}
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:99%;"
 |+ style="font-size: 75%;" |<span id='table-6'></span>'''Table. 6''' Convergence analysis of Example 6 using the IFD-IIM for the heat equation with a general discontinuous solution.
 |-
-|     0.99 @|c|cc|cc|cc|cc|
+! <math display="inline">N</math>
-| style="text-align: right;" |
+! <math display="inline">L_\infty </math>-norm
+!  Order
-|-
+! <math display="inline">L_2</math>-norm
-| <math display="inline">N</math>
+!  Order
-| <math display="inline">L_\infty </math>-norm
+! <math display="inline">L_\infty </math>-norm
-|  Order
+!  Order
-| <math display="inline">L_2</math>-norm
+! <math display="inline">L_2</math>-norm
-|  Order
+!  Order
-| <math display="inline">L_\infty </math>-norm
-|  Order
-| <math display="inline">L_2</math>-norm
-|  Order
 |-
 | 10
@@ Line 1,458: / Line 1,443: @@
 |  '''3.95'''
 |-
 |}