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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">Jed Brown <jed@jedbrown.org><br>
<b>Date: </b>Sunday, March 3, 2024 at 9:24 PM<br>
<b>To: </b>Zou, Ling <lzou@anl.gov>, petsc-users@mcs.anl.gov <petsc-users@mcs.anl.gov><br>
<b>Subject: </b>Re: [petsc-users] FW: 'Preconditioning' with lower-order method<o:p></o:p></span></p>
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<p class="MsoNormal" style="mso-line-height-alt:.75pt"><span style="font-size:1.0pt;color:white">One option is to form the preconditioner using the FV1 method, which is sparser and satisfies h-ellipticity, while using FV2 for the residual and (optionally) for
matrix-free operator application. FV1 is a highly diffusive method so in a sense,
<o:p></o:p></span></p>
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<pre style="white-space:pre-wrap"><span style="font-size:11.0pt;font-family:"Arial",sans-serif">One option is to form the preconditioner using the FV1 method, which is sparser and satisfies h-ellipticity, while using FV2 for the residual and (optionally) for matrix-free operator application.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;background:yellow;mso-highlight:yellow"><<< In terms of code implementation, this seems a bit tricky to me. Looks to me that I have to know exactly who is calling the residual function, if its MF operation, using FV2, while if finite-differencing for Jacobian, using FV1. Currently, I don’t know how to do it.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;background:yellow;mso-highlight:yellow">Another thing I’d like to mention is that the linear solver has never really been an issue. While the non-linear solver for the FV2 scheme often ‘stagnant’ at the first a couple of non-linear iteration [see the other email reply to Barry]. Seems to me, the additional nonlinearity from the TVD limiter causing difficulty to PETSc to find the attraction zone.</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">FV1 is a highly diffusive method so in a sense, it's much less faithful to the physics and (say, in the case of fluids) similar to a much lower-Reynolds number (if you use a modified equation analysis to work out the effective Reynolds number in the presence of the numerical diffusion).<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">It's good to put some thought into your choice of limiter. Note that intersection of second order and TVD methods leads to mandatory nonsmoothness (discontinuous derivatives). <o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;background:yellow;mso-highlight:yellow"><<< Yeah… I am afraid that the TVD limiter is the issue, so that’s the reason I’d try to use FV1 to bring the solution (hopefully) closer to the real solution so the non-linear solver has an easy job to do.</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">"Zou, Ling via petsc-users" <petsc-users@mcs.anl.gov> writes:<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Original email may have been sent to the incorrect place.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> See below.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> -Ling<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> From: Zou, Ling <lzou@anl.gov><o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Date: Sunday, March 3, 2024 at 10:34 AM<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> To: petsc-users <petsc-users-bounces@mcs.anl.gov><o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Subject: 'Preconditioning' with lower-order method<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Hi all,<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> I am solving a PDE system over a spatial domain. Numerical methods are:<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> * Finite volume method (both 1st and 2nd order implemented)<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> * BDF1 and BDF2 for time integration.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> What I have noticed is that 1st order FVM converges much faster than 2nd order FVM, regardless the time integration scheme. Well, not surprising since 2nd order FVM introduces additional non-linearity.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> I’m thinking about two possible ways to speed up 2nd order FVM, and would like to get some thoughts or community knowledge before jumping into code implementation.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Say, let the 2nd order FVM residual function be F2(x) = 0; and the 1st order FVM residual function be F1(x) = 0.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> 1. Option – 1, multi-step for each time step<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Step 1: solving F1(x) = 0 to obtain a temporary solution x1<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Step 2: feed x1 as an initial guess to solve F2(x) = 0 to obtain the final solution.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> [Not sure if gain any saving at all]<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> 1. Option -2, dynamically changing residual function F(x)<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> In pseudo code, would be something like.<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> snesFormFunction(SNES snes, Vec u, Vec f, void *)<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> {<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> if (snes.nl_it_no < 4) // 4 being arbitrary here<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> f = F1(u);<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> else<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> f = F2(u);<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> }<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> I know this might be a bit crazy since it may crash after switching residual function, still, any thoughts?<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> Best,<o:p></o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">><o:p> </o:p></span></pre>
<pre><span style="font-size:11.0pt;font-family:"Arial",sans-serif">> -Ling<o:p></o:p></span></pre>
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