Re: The relaxation method

["John S. Denker" <jsd@MONMOUTH.COM>]

Ludwik Kowalski wrote:
...
> 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?

You should use them to impose periodic boundary
conditions.  Do you know what that means?

> How should I relax marginal cells?

You shouldn't.  You should use them to impose periodic
boundary conditions.

> Please be as specific as
> possible; I am not very good in turning general principle,
> such conservation of Q or energy, into practical formulas.

I hear the request, but I'm going to turn it down
at least for today.  Please give it the old college
try.  See if you can figure it out for yourself.

1) Write down the definition of "periodic".  Stare at
it for a while.  See if that doesn't give you !!some!!
kind of an idea for an algorithm, and equation, a
really, really, really simple algorithm for calculating
the potential in neighboring copies of the universe,
given the potential in the main copy the universe.

2) Try clicking on one of the cells in question in my
spreadsheets (just outside adjacent to the main
universe) and see if you can reverse-engineer it.

3) If you really, really can't figure it out, ask
again.  Explain how you tried to figure it out and
why it didn't work.

==========================

> 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?

This is certainly not asking too much.
Fixing the potential _only_ at the inner electrodes
is a good idea.  There is no reason for not doing
it.  There are excellent reasons (pedagogical and
otherwise) for doing it:
 -- For starters, it results in a solution that is
    gauge-invariant.  Obviously a Good Thing (tm).
 -- It correctly represents the experiment as done
    on real resistor paper.  Again a Good Thing (tm).