[petsc-users] TS question 1: how to stop explicit methods because they do not use SNES(VI)?

[Jed Brown]

Ed Bueler <elbueler at alaska.edu> writes:

> Jed --
>
>> My point was that including sliding potentially adds another
>> stiff/algebraic term so whatever interface we choose better be able to
>> support at least two stiff terms.
>
> Yes, totally agreed w.r.t. the interface design.  (The DAE that you are
> solving in that case is stiff.)
>
> However, *my* observation about the current interface design is actually
> that it was never apparently tested with TS+SNESVI and is awkward in that
> case.

Certainly.

> IMHO the design for time-stepping on constrained problems should continue
> to allow use of SNESVI (in implicit and imex cases) if an LHSfunction (or
> similar) is provided, *and* have a mechanism where explicit time-solvers
> can use projection onto the bounds---conceivably this could be a
> trivialized SNESVI instance---if bounds are present.  This is because mere
> evaluation of LHSFunction and RHSFunction will not otherwise generate the
> "side effect" of enforcing the constraints which were provided by
> FormBounds() in using the SNESVI.
>
> In both the current and proposed designs in this thread, for implicit and
> imex methods the constraint enforcement is inside SNESVI *but* for explicit
> time-stepping it would be a user-written projection in TSPostStep().
> Furthermore the LHSFunction might have to be rewritten to allow
> non-feasible inputs just to support explicit multistage time-stepping.  Yuk.
>
> Note that there are VIs in L^2 and they *are* "projection onto the bounds"
> in the just-mentioned sense.  See section II.3 of Kinderlehrer and
> Stampacchia.  (Elliptic VIs in W^{1,p} are what one usually sees.)  The
> switch from VI in Sobolev space to VI in L^2 is exactly what you see when
> you semi-discretize these obstacle-like problems in time only and then
> compare implicit and explicit timestepping, respectively.
>
> So ya'll are having a good argument ... but I am worried about what I asked
> about anyway.

Should TS get explicit support for time-dependent VIs?

(Note that I joined the conversation late.)

>>> q_total = - Gamma H^{n+2} |grad s|^{n-1} grad s + V H
>> Is V really a prescribed function or the solution of an equation that
>> involves u?
>
> I know that the problem you want me to solve is a big DAE including both
> mass and momentum conservation.  (Do you really think that all this time I
> haven't noticed?)
>
> My interest is in good solvers for what you call the "local problem".  This
> local problem is the one that limits PISM hybrid runs *and* I think it is
> why Stokes is not really functional yet in anyone's time-stepping model.
>  (If they treat the time-stepping explicitly then they have the SIA-type
> time-step restriction kick in as a solver convergence problem, because the
> SIA tells the truth in the majority of the area where the bed is sticky.)
>  Thus my test case code does not waste time generating a real V (i.e.
> solving simultaneously for H and V at each time step, for the DAE you want
> me to solve ...).
>
>> For the SIA thickness equation, I think it would be interesting to
>> consider the conductivity tensor in the Newton linearization.
>>   d_u [-div (u^k |grad u|^{p-2} grad u)] * w
>>     = -div (k w^{k-1} |grad u|^{p-2} grad u)
>>       -div (u^k [|grad u|^{p-2} I + (p-2) |grad u|^{p-4} grad u \otimes
> grad u] grad w)
>> The first part is transport of the perturbation w.  The next is
>> diffusion of w with an anisotropic conductivity tensor.
>
> Yes I have considered it.  See http://dx.doi.org/10.1017/jog.2015.3, both
> the discussion of flux-splitting and the appendix on analytical jacobian.
> What do *you* get by considering this conductivity tensor?

Is it singular or degenerate apart from u=0 and grad u=0?  (I think this
can only happen for p=1.)  What is the maximum condition number of the
tensor?

Degeneracy is a common challenge in subsurface flows and AMG tends to do
alright, though it can be sensitive to the ability of the strength of
connection measure to diagnose anisotropy for the chosen discretization.

I guess my point is that at the level of a robust linear solve, I'm not
sure the degeneracy that you face is particularly different from that
faced in other applications.  If you want a robust nonlinear multigrid,
it's important to know what technology would be required for a robust
linear solve because you probably need at least that.

>>> So now I'll emphasize what I said before: it is *degenerate* p-bratu.
>> To my knowledge, "p-Bratu" is just something I invented to combine two
>> nonlinearities.  You don't have an e^u nonlinearity so there is no
>> Bratu-type nonlinearity.
>
> There *is* a term I want to treat explicitly, and it is *not* related to
> the stiffness etc., but to the coupling to the atmosphere.  (I said this
> before.)  Namely, in my form u_t - f(t,u)=g(t,u) the RHS g(t,u) is an
> elevation dependent mass balance term. Actually it can be (and might as
> well be) bratu i.e. exponentialish in the thickness.  Mostly it is out of
> my control because it is the altitude dependence of precipitation in the
> bottom few km of the atmosphere arising from coupling to atmosphere
> models.  My test case linearizes it a la a glaciologist, with an ELA and a
> lapse rate.
>
> By analogy, in other words, the time-dependent form of the
> I-know-Jed-invented-it p-bratu problem is pretty representative of my
> problem ... at least if you look in p>2 cases, add additional degeneracy to
> the diffusivity, make the zero-order term signed, use SNESVI (because the
> source term is signed and so a VI/NCP must be solved), and you add a
> sliding flux.  I was trying to work by analogy ...

Fair enough.
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