Title: A Hyperbolic Solver for Vacuum Axisymmetric Spacetimes: the method and weak wave tests
...work done in collaboration with David Garfinkle...
Outline:
Purpose: develop a vacuum axisymmetric solver using a combination of the
Geroch manifold of trajectories and a hyperbolic formulation.
Review Geroch split [David, here you might have some helpful ideas on the best
ways to do this without belaboring the point...]
Use Geroch split to get variables for hyperbolic formulation
Discuss the hyperbolic formulation, showing variables and
* flux friendly form
* primitive variable form
Show eigenvalues and eigenvectors, comment on use to construct
approximate characteristic variables in 'asymtotic' region
Numerical method: Lax-Wendroff second order accurate in space and
time
Show mesh, comment on axis bc and outer bc
Tests for weak waves:
test using non-time symmetric initial data
show surface plots of some variables for exact
solution and computed solution.
comment on remaining problems [axis, probably?]
Next steps:
evolve stronger Brill waves
add shift vector
add in twist
redo solver using more accurate methods
Subject: manifold of trajectories
Hi Comer,
I have sent you the last version of the code. The agreement
between krr and ckrr is pretty good in the "eyeball norm" except at the
axis.
Thanks for asking me to chair a session. I am glad to do so.
Your talk outline looks good to me. As for how to explain the
Geroch manifold of trajectories you might try something like the
following: in the case of no rotation (which is the only case that we
treat so far) the manifold of trajectories is equivalent to a surface of
constant \phi. So you could just say that in an axisymmetric spacetime
knowing what happens on a surface of constant \phi is equivalent to
knowing what happens in the whole spacetime. (that should be good
enough for a 15 minute talk). Then you can just mention that with
rotation something analogous (but a bit more complicated) can be done.
--David