Aslan,
There is, however, one important drawback of the multistep methods (including
BDF of order 2<k<6 that I am looking at): they require solution transfer from
more than one previous steps compared to just one in one-step methods
(Runge-Kutta, theta scheme). Therefore, if the mesh is continuously
refined/coarsened during the simulation, this will lead to increase in
computational expenses due to necessity of interpolation of several solutions
onto the new grid.
The way you would implement this is that every time you change the mesh, you
transfer all solution vectors from the previous to the new mesh. The old mesh,
and all solution vectors defined on it, then cease to exist. Everything only
exists on the new mesh. For all practical purposes, at any given time of the
time stepping method, you only ever work on one mesh and it looks like there
has never been another mesh. The only time when you deal with two meshes is
during the brief period where you use SolutionTransfer.
Now, transferring solution vectors is not cheap, but it's also not expensive
compared to, for example, assembling and solving a linear system. That's
particularly true because you don't typically adapt the mesh very frequently.
For most problems, I would assume that adapting the mesh every 10 or 20 time
steps is perfectly fine. For example, in our "big" code, ASPECT, the default
is every 15 time steps, and I don't know that anyone would have ever changed
that default in their applications. In any case, if you amortize the cost of
solution transfer over 10 or 20 time steps, then it's pretty much negligible.
I would say that's true regardless of whether you transfer one, or ten
solution vectors.
Best
Wolfgang
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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