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

[Matthew Knepley]

On Mon, Apr 6, 2020 at 2:06 PM Alexander B Prescott <
alexprescott at email.arizona.edu> wrote:

> 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.
>

My understanding is that this is a shortcoming of the model, not the
solver. However, I am Cc'ing Nathan since he knows
about these models.

  Thanks,

    Matt


> 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
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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