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 <[email protected]> on behalf of Alexander B
Prescott <[email protected]>
Date: Monday, April 6, 2020 at 1:06 PM
To: PETSc <[email protected]>
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
[email protected]<mailto:[email protected]>
PhD Candidate, The University of Arizona
Department of Geosciences
1040 E. 4th Street
Tucson, AZ, 85721
