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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, 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).
"Zou, Ling via petsc-users" <petsc-users@mcs.anl.gov> writes:
> Original email may have been sent to the incorrect place.
> See below.
>
> -Ling
>
> From: Zou, Ling <l...@anl.gov>
> Date: Sunday, March 3, 2024 at 10:34 AM
> To: petsc-users <petsc-users-boun...@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
