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

[Zou, Ling]

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<mailto: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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