Re: The relaxation method

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

John Mallinckrodt wrote:

> Having nothing better to do than several stack of papers to
> grade and looking for a way to avoid that, I dusted off the old
> relaxation method spreadsheet, used it to calculate the potentials
> for Ludwik's conducting sheet problem, and got results that
> can be seen at
>
>  <http://www.csupomona.edu/~ajm/special/condsheet2d.gif>
>
> and
>
>  <http://www.csupomona.edu/~ajm/special/condsheet3d.gif>
>
> I simulated 1/4 of the sheet using the same dimensions (14 units
> in the direction of the dipole by 10 units) and the same pole
> diameter (1 unit) that I used with my image charge solution.  In
> both cases I used 20 equally spaced equipotentials between the
> midplane and the positive pole. The results match in detail to the
> resolution I can discern.
>
> Just a reminder that the image charge solution can be seen at
>
>  <http://www.csupomona.edu/~ajm/special/condsheet2.pdf>

I will also try to implement the relaxation method (time permitting).
Why do I want to do this when the problem has already been
solved by JohnM? Because I want to learn physics.

I see that John's equipotentials are everywhere perpendicular
to the paper borders. How did you relax marginal cells (those
which have only three or two neighbors), John, to get this?

I was not able to figure out how to deal with marginal cells
from looking at what was posted by JohnD. He shows the
algorithm for the inner cells which have four neighbors but
not for the cells with three or two neighbors. The answer is
probably hidden in this:

> > Thirdly, just outside the edge of the universe, there
> > is another layer of cells. These implement what I call
> > the boundary condition for the universe itself. ..."

That is where I am confused. Suppose I do create a layer of
cells adjacent to the Pasco sheet area. These are artificial
neighbors of marginal cells. What should I do with them?
How should I relax marginal cells? Please be as specific as
possible; I am not very good in turning general principle,
such conservation of Q or energy, into practical formulas.
Ludwik Kowalski
P.S.
Last year I did not have this problem; I simply imposed
a constant zero potential on every marginal cell. Margins
were far away from the region in which I was interested;
they were the boundary of the universe. But I do not want
to impose any apriori potentials on marginal cells; I want to
learn what actually happens at marginal cells, according to
the relaxation algorithm. The only things I want to impose
are, for example, -15 V and +15 V at the inner cells
representing silver-painted circles. Am I asking too much?

My impression was that to solve the Laplace equation one
must first impose the values of V on all surrounding cells.
But JohnD, if I understand him correctly, tells us that this
is not necessarily true. How should I relax marginal cells?
I plan to program in True Basic, not in Excel and not in
Maple. Why? Because it did become my tool; I am more
comfortable with True Basic. [By the way, the student
version of True Basic, costing about $20 (including the
little tutorial book), no longer has the limit on the number
of lines; it is as powerful as the professional version. For
more information call 1-800-TR-BASIC.]
Ludwik Kowalski