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

[Zou, Ling]

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
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