I would like to know how one can do a time-integration at a fixed
location on the grid, in a multi-grid situation.

For example, right now I have a piece of auxiliary code where i
compute a new field variable NEW at each time step, from which I want
its time integral W[7].  For a time integration, I simply cumulate the
updated value of NEW at that grid location, e.g.
W(7,I,J) = NEW+W(7,I,J)
How is that impacted by the fact that I have multiple grid levels and
the grid gets refined/de-refined as time advances   Seems to work, but
are there any issues?
Thanks,
matei

Matei,

Yes. You have to ask yourself what are AMR_SOL's prolongation and
restriction operators doing to your NEW quantity. The prolongation
is the interpolation done when putting down a new fine patch that
can't lift its solution from a previous fine patch. The restriction
is the averaging done on a fine patch when projecting its solution
onto the underlying coarse patch.

The default restriction operator assumes the quantities are in conserved
form and in effect computes a mass weighted average. The default
prolongation operator performs a MUSCL type reconstruction. This is
sympathetic to the types of shock-capturing schemes used by AMR_SOL,
but it results in a stepped interpolation that might cause grief
depending on what you're doing.

It is possible to overload the restriction and prolongation operators
via the patch-integrator. But then you're getting down to the bowels
of the machinery, with all the nitty gritty problems that entails.

James