[petsc-users] [EXT]Re: Discontinuities in the Jacobian matrix for nonlinear problem

[Alexander B Prescott]

Thank you Ling, that's an interesting idea. Bending the physics like this
would be okay for my purposes, I'll give it a look.

Best,
Alexander

On Mon, Apr 6, 2020 at 11:56 AM Zou, Ling <lzou at anl.gov> wrote:

> *External Email*
>
> What about ‘bending’ the physics a bit?
>
>
>
> Q_i-1/2 proportional to sqrt(x[i-1] + z[i-1] - (x[i] + z[i])),
>       if             x[i-1]+z[i-1]  >  x[i]+z[i] + eps
>
> Q_i-1/2 proportional to -1.0*sqrt(x[i] + z[i] - (x[i-1] + z[i-1])),
>    if            x[i]+z[i]  >  x[i-1]+z[i-1] + eps
>
> Q_i-1/2 proportional to x[i-1] + z[i-1] - (x[i] +
> z[i])                               in between
>
>
>
> in which, eps is a very small positive number.
>
>
>
> -Ling
>
>
>
>
>
> *From: *petsc-users <petsc-users-bounces at mcs.anl.gov> on behalf of
> Alexander B Prescott <alexprescott at email.arizona.edu>
> *Date: *Monday, April 6, 2020 at 1:06 PM
> *To: *PETSc <petsc-users at mcs.anl.gov>
> *Subject: *[petsc-users] Discontinuities in the Jacobian matrix for
> nonlinear problem
>
>
>
> Hello,
>
>
>
> The non-linear boundary-value problem I am applying PETSc to is a
> relatively simple steady-state flow routing algorithm based on the
> continuity equation, such that Div(Q) = 0 everywhere (Q=discharge). I use a
> finite volume approach to calculate flow between nodes, with Q calculated
> as a piecewise smooth function of the local flow depth and the
> water-surface slope. In 1D, the residual is calculated as R(x_i)=Q_i-1/2 -
> Q_i+1/2.
>
> For example, Q_i-1/2 at x[i]:
>
> Q_i-1/2 proportional to sqrt(x[i-1] + z[i-1] - (x[i] + z[i])),
>   if  x[i-1]+z[i-1]  >  x[i]+z[i]
>
> Q_i-1/2 proportional to -1.0*sqrt(x[i] + z[i] - (x[i-1] + z[i-1])),
> if         x[i]+z[i]  >  x[i-1]+z[i-1]
>
>
>
> Where z[i] is local topography and doesn't change over the iterations, and
> Q_i+1/2 is computed analogously. So the residual derivatives with respect
> to x[i-1], x[i] and x[i+1] are not continuous when the water-surface slope
> = 0.
>
>
>
> Are there intelligent ways to handle this problem? My 1D trial runs
> naively fix any zero-valued water-surface slopes to a small non-zero
> positive value (e.g. 1e-12). Solver convergence has been mixed and highly
> dependent on the initial guess. So far, FAS with QN coarse solver has been
> the most robust.
>
>
>
> Restricting x[i] to be non-negative is a separate issue, to which I have
> applied the SNES_VI solvers. They perform modestly but have been less
> robust.
>
>
>
> Best,
>
> Alexander
>
>
>
>
>
>
> --
>
> Alexander Prescott
>
> alexprescott at email.arizona.edu
>
> PhD Candidate, The University of Arizona
>
> Department of Geosciences
>
> 1040 E. 4th Street
>
> Tucson, AZ, 85721
>


-- 
Alexander Prescott
alexprescott at email.arizona.edu
PhD Candidate, The University of Arizona
Department of Geosciences
1040 E. 4th Street
Tucson, AZ, 85721
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