[overture] Re: Poisson equation in spherical coordinates
- From: Bill Henshaw <henshaw@xxxxxxxx>
- To: overture@xxxxxxxxxxxxx
- Date: Thu, 23 Jul 2009 20:53:22 -0700
Dear Slava,
I am used to (theta,phi) having the reverse meaning but apparently
your usage is the ISO standard!?
Your condition is a generalized periodicity condition and I hope
that you have another equation to use since it doesn't
seem you can apply your condition at all values of phi without
getting duplicate equations (?) Perhaps you can add a ghost point.
In any case see the example Overture/tests/tcmge.C (in v23) on how
to fill in the coefficient matrix with any equations that
you like. There is no high-level function that will do
what you are asking.
Regards,
Bill.
Viacheslav Merkin wrote:
Dear Bill,
I am solving a Poisson equation in spherical coordinates using Ogen.
Physically the equation is written on a spherical shell; I'm solving it
on a rectangular grid (theta,phi), where theta is the polar angle and
phi is the azimuthal angle. I have been trying different boundary
conditions at the pole and even had a conversation with you about it a
fairly long while ago. I've used a boundary condition since then which
worked well but I have doubts now about its justification. Upon doing
proper Taylor expansion around the pole, I realized that (surprise!) the
electric field component E_theta in the vicinity of the pole is given by
A*sin(phi)+B*cos(phi) to zeroth order in theta, which says that the
electric field is uniform. Considering next terms in the expansion it is
possible to derive a linear relationship between A and B, but it is not
necessary to set up a boundary condition. From the uniformity of the
field it follows that a good boundary condition to use is
E_theta(theta=0,phi) = - E_theta(theta=0,PI+phi), where
E_theta=-d_Psi/d_theta and Psi is the potential. The above boundary
condition ensures the smoothness of the electric field over the pole.
So, here is my question: is it possible to "easily" set this boundary
condition in Overture without modifying the coefficient matrix, and if
not, how do I go about modifying the coefficient matrix?
Thank you very much in advance,
Slava Merkin
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Viacheslav Merkin
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Senior Research Associate
Astronomy Department and
Center for Integrated Space Weather Modeling
Boston University
e-mail: vgm at bu.edu
phone: (617) 358-3441
fax: (617) 358-3242
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