[petsc-users] FW: 'Preconditioning' with lower-order method

[Zou, Ling]

From: Jed Brown <jed at jedbrown.org>
Date: Sunday, March 3, 2024 at 9:24 PM
To: Zou, Ling <lzou at anl.gov>, petsc-users at mcs.anl.gov <petsc-users at mcs.anl.gov>
Subject: Re: [petsc-users] FW: 'Preconditioning' with lower-order method
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,
ZjQcmQRYFpfptBannerStart
This Message Is From an External Sender
This message came from outside your organization.

ZjQcmQRYFpfptBannerEnd

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.



<<< 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.

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.



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).



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).



<<< 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.



"Zou, Ling via petsc-users" <petsc-users at mcs.anl.gov> writes:



> Original email may have been sent to the incorrect place.

> See below.

>

> -Ling

>

> From: Zou, Ling <lzou at anl.gov>

> Date: Sunday, March 3, 2024 at 10:34 AM

> To: petsc-users <petsc-users-bounces at mcs.anl.gov>

> Subject: 'Preconditioning' with lower-order method

> Hi all,

>

> I am solving a PDE system over a spatial domain. Numerical methods are:

>

>   *   Finite volume method (both 1st and 2nd order implemented)

>   *   BDF1 and BDF2 for time integration.

> 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.

>

> 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.

>

> Say, let the 2nd order FVM residual function be F2(x) = 0; and the 1st order FVM residual function be F1(x) = 0.

>

>   1.  Option – 1, multi-step for each time step

> Step 1: solving F1(x) = 0 to obtain a temporary solution x1

> Step 2: feed x1 as an initial guess to solve F2(x) = 0 to obtain the final solution.

> [Not sure if gain any saving at all]

>

>

>   1.  Option -2, dynamically changing residual function F(x)

>

> In pseudo code, would be something like.

>

>

>

> snesFormFunction(SNES snes, Vec u, Vec f, void *)

>

> {

>

>   if (snes.nl_it_no < 4) // 4 being arbitrary here

>

>     f = F1(u);

>

>   else

>

>     f = F2(u);

>

> }

>

>

>

> I know this might be a bit crazy since it may crash after switching residual function, still, any thoughts?

>

> Best,

>

> -Ling
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20240304/df35bf43/attachment.html>