Hi Jie,
> Thank you for your hints on my previous question. Now I have
> extended step-57 to be time-dependent. I added a u,t term which is
> approximated using backward Euler, and all the remaining terms are
> treated implicitly. Newton iteration is still used at every time step.
> The system becomes
> (A + M/dt)U = F - (\delta v, (u^{k+1} - u^k)/dt)_\Omega, BU = P,
> which is similar to what we have in step-57 except the LHS(0, 0) and
> RHS(0) are changed.
>
> I'm still using the same preconditioner developed in step-57. So far
> it works the simple case of lid-driven cavity flow. My questions are:
> 1. It doesn't make much sense to keep using the preconditioner
> developed for steady NSE here, are there any simple ways to adapt it
> for time-dependent problems?
You should check the literature. In case your time step is not too
large, the mass matrix will give a significant contribution to the
system matrix. This is good because one knows how to approximate the
Schur complement for a velocity mass matrix - it is simply a pressure
Poisson matrix, with Dirichlet conditions on that operator where the
velocity has Neumann and Neumann conditions where you specify the flow.
There are several variants of preconditioners in the literature. One is
Cahouet-Chabard 1988 that simply uses the inverse Poisson matrix and the
inverse mass matrix to take the mass and viscous contributions in
velocity into account. I have used it in a recent publication:
https://doi.org/10.1177/1094342016671790 - check also literature cited
there, this mailing list is not the right place to ask for a literature
review.
Then, there are also pressure reaction-convection-diffusion operators
that also take convection into account. I have not seen them a lot for
the full block system in the time-dependent case, probably because the
range where they are really better than Cahouet-Chabard yet the Schur
complement approximation stays good enough is quite narrow - especially
once you also want to have good yet cheap approximation of the velocity
matrix.
> 2. Instead of working on better preconditioners, can I simply treat
> the convection term explicitly (using the value at last time step)?
> The reason is that doing so would make the LHS of the equation
> symmetric (I think), which should be easy to solve. This seems to be
> much easier.
Yes, this is Cahouet-Chabard and indeed very efficiently done. But you
get a CFL limit which may or may not be limiting for you.
Best,
Martin
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