[overture] Poisson equation in spherical coordinates
- From: Viacheslav Merkin <vgm@xxxxxx>
- To: overture@xxxxxxxxxxxxx
- Date: Thu, 23 Jul 2009 09:49:18 -0400
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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