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Approximation Schemes for convective term - structured grids - Summary of Discretizations Schemes and examples - Revision history
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Michail: /* Example code for solving Smith-Hutton problem */
2012-04-21T05:56:18Z
<p><span class="autocomment">Example code for solving Smith-Hutton problem</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 05:56, 21 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Sample code for solving Smith-Hutton test - Fortran 90]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Sample code for solving Smith-Hutton test - Fortran 90]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>It's <del class="diffchange diffchange-inline">a </del>results, obtained using this code ('''UDS''' and '''HLPA''' schemes)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>It's results, obtained using this code ('''UDS''' and '''HLPA''' schemes)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
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Michail
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Michail: /* Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem) */
2012-04-20T19:01:49Z
<p><span class="autocomment">Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 19:01, 20 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The two parameters which define the scalar field are the Peclet number, which specifies the diffusivity of the problem, and <math>\boldsymbol{\alpha}</math> which is a parameter that defines the sharpness of the inlet profile.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The two parameters which define the scalar field are the Peclet number, which specifies the diffusivity of the problem, and <math>\boldsymbol{\alpha}</math> which is a parameter that defines the sharpness of the inlet profile.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Below it's cleary seen the numerical diffusion impact, comparing the contour fields obtaining using the UDS and HLPA. A bit later we shall place here a solution gained with QUICK scheme, and it will be seen the osscilations.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:NM_convectionschemes_struct_grids_Smith_hutton_Countor_HLPA._probe_01.jpg]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:NM_convectionschemes_struct_grids_Smith_hutton_Countor_HLPA._probe_01.jpg]]</div></td></tr>
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Michail
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Michail: /* Example code for solving Smith-Hutton problem */
2012-04-20T19:01:20Z
<p><span class="autocomment">Example code for solving Smith-Hutton problem</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It's a results, obtained using this code ('''UDS''' and '''HLPA''' schemes)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It's a results, obtained using this code ('''UDS''' and '''HLPA''' schemes)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Below it's cleary seen the numerical diffusion impact, comparing the contour fields obtaining using the UDS and HLPA. A bit later we shall place here a solution gained with QUICK scheme, and it will be seen the osscilations.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td></tr>
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</table>
Michail
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Michail: /* Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem) */
2012-04-20T18:46:57Z
<p><span class="autocomment">Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 18:46, 20 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:NM_convectionschemes_struct_grids_Smith_hutton_Countor_HLPA_02_probe_03.jpg]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:NM_convectionschemes_struct_grids_Smith_hutton_Countor_HLPA_02_probe_03.jpg]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Below is the outlet profiles for solution of Smith-Hutton test. Clearly seen the osscilation for QUICK scheme and it can be noticed that HLPA free of its</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Below is the outlet profiles for solution of Smith-Hutton test. Clearly seen <ins class="diffchange diffchange-inline">the diffusivity of UDS, </ins>the osscilation for QUICK scheme and it can be noticed that HLPA free of its <ins class="diffchange diffchange-inline">with the same accuracy.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:Outlet_profiles.JPG]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:Outlet_profiles.JPG]]</div></td></tr>
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Michail
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Michail: /* Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem) */
2012-04-20T18:45:19Z
<p><span class="autocomment">Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 18:45, 20 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:NM_convectionschemes_struct_grids_Smith_hutton_Countor_HLPA_02_probe_03.jpg]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:NM_convectionschemes_struct_grids_Smith_hutton_Countor_HLPA_02_probe_03.jpg]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Below is the outlet profiles for solution of Smith-Hutton test. Clearly seen the osscilation for QUICK scheme and it can be noticed that HLPA free of its</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[[Image:Outlet_profiles.JPG]]</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Square Lid-driven cavity flow ===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Square Lid-driven cavity flow ===</div></td></tr>
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Michail
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Michail: /* Discretizations Schemes - Estimation of order - Richardson extrapolation */
2012-04-08T11:48:59Z
<p><span class="autocomment">Discretizations Schemes - Estimation of order - Richardson extrapolation</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 11:48, 8 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="100%"><tr><td></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="100%"><tr><td></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>r= \left(h_{3}/<del class="diffchange diffchange-inline">h</del>{2} \right) = \left( h_{2}/h{1} \right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>r= \left(h_{3}/<ins class="diffchange diffchange-inline">h_</ins>{2} \right) = \left( h_{2}/h{1} \right)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(6)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(6)</td></tr></table></div></td></tr>
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Michail
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Michail: /* Discretizations Schemes - Estimation of order - Richardson extrapolation */
2012-04-08T11:48:26Z
<p><span class="autocomment">Discretizations Schemes - Estimation of order - Richardson extrapolation</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 11:48, 8 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(5)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(5)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><table width="100%"><tr><td></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\widetilde{p} = ln \left( \frac{\varphi_{h_{3}} - \varphi_{h_{2}}}{\varphi_{h_{2}}-\varphi_{h_{1}} } \right) / ln \left( r \right)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></td><td width="5%">(6)</td></tr></table></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><table width="100%"><tr><td></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">r= \left(h_{3}/h{2} \right) = \left( h_{2}/h{1} \right)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></td><td width="5%">(6)</td></tr></table></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Selection advice ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Selection advice ==</div></td></tr>
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Michail
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Michail: /* Discretizations Schemes - Estimation of order - Richardson extrapolation */
2012-04-08T11:42:55Z
<p><span class="autocomment">Discretizations Schemes - Estimation of order - Richardson extrapolation</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 11:42, 8 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(3)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(3)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(4)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(4)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><table width="100%"><tr><td></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\left\| \varphi_{exact} - \varphi_{h_{1}} \right\| \cong \left| \left[ \frac{\varphi_{h_{2}} - \varphi_{h_{1}}}{h^{p}_{2} - h^{p}_{1}} \right]h^{p}_{1} \right| + O \left(h^{p+1}_{1} \right) + O \left(h^{p+1}_{2} \right)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></td><td width="5%">(5)</td></tr></table></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Selection advice ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Selection advice ==</div></td></tr>
</table>
Michail
https://www.cfd-online.com/W/index.php?title=Approximation_Schemes_for_convective_term_-_structured_grids_-_Summary_of_Discretizations_Schemes_and_examples&diff=14127&oldid=prev
Michail: /* Discretizations Schemes - Estimation of order - Richardson extrapolation */
2012-04-08T11:06:52Z
<p><span class="autocomment">Discretizations Schemes - Estimation of order - Richardson extrapolation</span></p>
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<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 11:06, 8 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(3)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(3)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><table width="100%"><tr><td></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\varphi_{exact} = \left[ \frac{h^{p}_{2}\varphi_{h_{1}} - h^{p}_{1}\varphi_{h_{2}} }{h^{p}_{2} - h^{p}_{1}} \right] + O \left(h^{p+1}_{1} \right) + O \left(h^{p+1}_{2} \right)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></td><td width="5%">(4)</td></tr></table></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Selection advice ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Selection advice ==</div></td></tr>
</table>
Michail
https://www.cfd-online.com/W/index.php?title=Approximation_Schemes_for_convective_term_-_structured_grids_-_Summary_of_Discretizations_Schemes_and_examples&diff=14126&oldid=prev
Michail: /* Discretizations Schemes - Estimation of order - Richardson extrapolation */
2012-04-08T10:57:05Z
<p><span class="autocomment">Discretizations Schemes - Estimation of order - Richardson extrapolation</span></p>
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<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 10:57, 8 April 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\alpha = \left[ \frac{<del class="diffchange diffchange-inline">u_</del>{h_{1}} - <del class="diffchange diffchange-inline">u_</del>{h_{2}}}{ h^{p}_{2} - h^{p}_{1}} \right] + O \left(h^{p+1}_{1} \right) + O \left(h^{p+1}_{2} \right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\alpha = \left[ \frac{<ins class="diffchange diffchange-inline">\varphi_</ins>{h_{1}} - <ins class="diffchange diffchange-inline">\varphi_</ins>{h_{2}}}{ h^{p}_{2} - h^{p}_{1}} \right] + O \left(h^{p+1}_{1} \right) + O \left(h^{p+1}_{2} \right)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(3)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(3)</td></tr></table></div></td></tr>
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Michail