James, can you perhaps send the FipY code for one of these equations, or a
similar variant of it?
Thanks!
Sergio
Sergio Manzetti
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From: "James Pringle"
To: "fipy"
Sent: Wednesday, May 31, 2017 10:50:11 PM
Subject: Re: solvers for elliptic equations with dominant lower order terms
I may have been unclear. This is a single PDE in the variable eta. A(x,y),
B(x,y) and C(x,y) are coefficients of the PDE. They are indeed separate, and
fipy does a nice job of solving this sort of equation in many limits. I am
having problems when the lower order parts of the equation ( B(x,y)*
eta_x+C(x,y)*eta_y) dominate the solution and the characteristics defined by B
and C form closed loops.
To understand this better, imagine the equation as a steady state fluid flow
problem, with the A(x,y)*(eta_xx+eta_yy) representing diffusion (and to be
correct, it should be written "Del . (A(x,y) Grad eta)"), and the low order
terms representing advection, with B and C being the velocities doing the
advection. When the diffusion is negligible, eta should be constant along the
flow lines (characteristics) defined by B and C. But if those flow line form a
closed loop, the diffusive term must determine the overall solution inside the
loop...
Thanks,
Jamie
On Wed, May 31, 2017 at 3:32 PM, Sergio Manzetti < [
mailto:sergio.manze...@fjordforsk.no | sergio.manze...@fjordforsk.no ] > wrote:
James, are A(x,y), B(x,y) and C(x,y) three different variable functions for the
PDE?
FIPY can solve at least when these three are equal. If it solves when they
differ, I am not sure.
Sergio
Sergio Manzetti
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From: "James Pringle" < [ mailto:jprin...@unh.edu | jprin...@unh.edu ] >
To: "fipy" < [ mailto:fipy@nist.gov | fipy@nist.gov ] >
Sent: Wednesday, May 31, 2017 9:31:51 PM
Subject: solvers for elliptic equations with dominant lower order terms
Dear all --
I need to solve a second order PDE in 2D of the form
BQ_BEGIN
A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0
This is an advective diffusive balance, with B and C representing the advective
velocities.
BQ_BEGIN
BQ_END
Over much of the domain the low order terms dominate. For some geometries, the
default solver I am using (SciPy LinearLUSolver) is having trouble find an
answer (especially when the characteristics defined by B and C form loops .
Does anyone have recommendations for how to choose a better solver? (I have
blindly tried all the ones in SciPy, the ones besides LinearLUSolver do worse).
And are there any hints on how to use preconditioners? There is discussion of
preconditioners and Trilinnos, but I can't find hints on how one would choose
to use them and when they might help.
Pointers to the broader literature are welcome!
Thank you,
Jamie Pringle
___
fipy m
