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<p class="MsoNormal"><span style="font-size:11.0pt">Original email may have been sent to the incorrect place.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">See below.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">-Ling<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal" style="margin-bottom:12.0pt"><b><span style="font-size:12.0pt;font-family:"Aptos",sans-serif;color:black">From:
</span></b><span style="font-size:12.0pt;font-family:"Aptos",sans-serif;color:black">Zou, Ling <lzou@anl.gov><br>
<b>Date: </b>Sunday, March 3, 2024 at 10:34 AM<br>
<b>To: </b>petsc-users <petsc-users-bounces@mcs.anl.gov><br>
<b>Subject: </b>'Preconditioning' with lower-order method<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt">Hi all,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">I am solving a PDE system over a spatial domain. Numerical methods are:<o:p></o:p></span></p>
<ul style="margin-top:0in" type="disc">
<li class="MsoListParagraph" style="margin-left:0in;mso-list:l0 level1 lfo1"><span style="font-size:11.0pt">Finite volume method (both 1<sup>st</sup> and 2<sup>nd</sup> order implemented)<o:p></o:p></span></li><li class="MsoListParagraph" style="margin-left:0in;mso-list:l0 level1 lfo1"><span style="font-size:11.0pt">BDF1 and BDF2 for time integration.<o:p></o:p></span></li></ul>
<p class="MsoNormal"><span style="font-size:11.0pt">What I have noticed is that 1<sup>st</sup> order FVM converges much faster than 2<sup>nd</sup> order FVM, regardless the time integration scheme. Well, not surprising since 2<sup>nd</sup> order FVM introduces
additional non-linearity.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">I’m thinking about two possible ways to speed up 2<sup>nd</sup> order FVM, and would like to get some thoughts or community knowledge before jumping into code implementation.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Say, let the 2<sup>nd</sup> order FVM residual function be
<b><i>F</i></b><sub>2</sub>(<b><i>x</i></b>) = 0; and the 1<sup>st</sup> order FVM residual function be
<b><i>F</i></b><sub>1</sub>(<b><i>x</i></b>) = 0.<o:p></o:p></span></p>
<ol style="margin-top:0in" start="1" type="1">
<li class="MsoListParagraph" style="margin-left:0in;mso-list:l1 level1 lfo2"><span style="font-size:11.0pt">Option – 1, multi-step for each time step<o:p></o:p></span></li></ol>
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:11.0pt">Step 1: solving
<b><i>F</i></b><sub>1</sub>(<b><i>x</i></b>) = 0 to obtain a temporary solution <b>
<i>x</i></b><sub>1</sub><o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:11.0pt">Step 2: feed
<b><i>x</i></b><sub>1</sub> as an initial guess to solve <b><i>F</i></b><sub>2</sub>(<b><i>x</i></b>) = 0 to obtain the final solution.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:11.0pt">[Not sure if gain any saving at all]<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<ol style="margin-top:0in" start="2" type="1">
<li class="MsoListParagraph" style="margin-left:0in;mso-list:l2 level1 lfo3"><span style="font-size:11.0pt">Option -2, dynamically changing residual function F(x)<o:p></o:p></span></li></ol>
<p class="MsoListParagraph"><span style="font-size:11.0pt">In pseudo code, would be something like.<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco">snesFormFunction(SNES snes, Vec u, Vec f, void *)</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco">{</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco"> if (snes.nl_it_no < 4) // 4 being arbitrary here</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco"> f = F1(u);</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco"> else</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco"> f = F2(u);</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco">}</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-family:Monaco"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">I know this might be a bit crazy since it may crash after switching residual function, still, any thoughts?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:Monaco"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:Monaco">Best,</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:Monaco"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:Monaco">-Ling</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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