On 12/7/20 6:19 AM, blais...@gmail.com wrote:
How come you get away with only assembling the matrix every 3-5 time steps?
Are you treating advection explicitly?
We get off by assembling the matrix every N time-steps for two main reasons:
- We use SDIRK which has a constant diagonal for all stages, hence this part
of the matrix does not change significantly
- We are willing to sacrifice one non-linear iteration there and there, at the
cost of less assembly. The idea is that we use an exact jacobian when we
assemble the matrix, but then we let it become innacurate as we continue
iterating, while just updating the rhs. This is not a good approach when the
time-step is large, but when the time-step is small (say CFL<1), it really
does not affect significantly the number of newton iterations. Consequently,
we can update the jacobian matrix every 3 to 5 time-steps or so. This is just
an indication, generally we update it as soon as a non-linear step has not
lead to a decrease of more than 10x of the residual, meaning that our Newton
method is becoming inefficient.
Nice approach!
I'm also curious about what the matrix you use looks like. It must be
something like
A B
B^T C
What terms are in A,B,C?
It has exactly this shape. The full weak form and some results are detailed in
: https://www.sciencedirect.com/science/article/pii/S2352711020302922
A is momentum + SUPG
C is the PSPG block. In essence it looks very much like a stiffness matrix
weighted by the stabilization parameter. Hence, I really don't think it is
"well conditioned".
Timo and Martin have already answered the question that underlaid why I asked
about the block structure. In practice, you can't get good preconditioners if
you don't exploit what a matrix corresponds to. ILU doesn't work well for this
reason, but if you can decompose a matrix into its blocks, you can get optimal
preconditioners for each block (say, the top left A block, given what it
represents -- if you resolve the length scales of advection, i.e., if your
cell Peclet number is less than 1, then it's essentially an elliptic operator
for which you can apply multigrid methods). Then you build your overall
preconditioner from preconditioners for each block separately.
I second the suggestion of the book by Elman, Silvester, and Wathen. I would
also point you to step-22. The implementation in the actual program is more
for illustrative purposes of what one can do, but I would suggest you
specifically read the section in the "Possibilities for extensions" that
discusses the Silvester/Wathen preconditioner which is probably close to
optimal. (It uses a mass matrix for the Schur complement, and that can be done
better using the BTFB method, but I think it points you in the right direction).
Completely separately, one of the things Timo and I played around with in
ASPECT a long time ago is whether to make some terms in the equation explicit
or implicit. We had originally made advection explicit, because that then
leads to a symmetric system matrix and we thought that that would be a more
efficient way to solve the linear system. And that's true, symmetric systems
are faster to solve. But it required us to have a substantially smaller time
step, and in the end we realized that the overall time spent is (time per time
step) * (number of time steps) and that by doing everything implicit, we could
increase the time step size by 10x whereas the cost for solving the linear
systems only increased by 2x, so the made the whole simulation cheaper by 5x.
I think that from your description you're already doing everything implicitly,
but it's worth thinking about these kinds of trade-offs anyway if you
currently have anything that limits your time step size.
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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