**Frontiers
in Numerical Gravitational Astrophysics**

2^{nd}
Course of the International School
on Astrophysical Relativity

**«John Archibald Wheeler»**

Scientific Summary

The meeting has the format of an advanced school, where high level scientists speak of their work in the field of the study of gravitating systems in Astrophysics, both on the classic and relativistic side, keeping their lectures at a level clearly understandable to graduate students.

The objective of the course is to learn from Newtonian physics and to
extend this knowledge to Relativistic computational astrophysics.
Physical examples in the course will begin with computational Newtonian
dynamics both for single stars (initial instability and core formation
through accretion, final evolutionary phases and collapse, supernovae)
and stellar systems (open and globular clusters, galaxies). We will
then move to relativistic computational physics. Every Session of
the School will face a topic on two parallel (Newtonian and
Relativistic) points of view.

Topics

The initial collapse and fragmentation of a molecular cloud leads to star formation. Physics is higly non-linear, and gravity is strongly coupled to microphysics. General Relativity is usually not considered relevant in this context, because the fate of a massive gas cloud, almost free falling dissipatively, is not commonly studied. But this may be one of the origins of black holes.

With
regard to star collapse and explosion, the methods in the
Newtonian and the Relativistic versions of discretization are very
similar. However, differences arise between the two Newtonian and the
Relativistic approaches, mainly because of the coordinate invariance of
General Relativity.

The dynamics of stellar systems, even in
presence of massive black
holes in their centers, usually relies on integration of Newtonian
equations of *N* interacting objects. The methods are mainly
devoted to both numerical accuracy and time consumption reduction; in
any case, attention should be given to a proper physical treatment
(relativity) when dealing with the phenomenon of mass accretion around
the singularity. This is indeed a still open problem,
fundamental into the definition of supermassive black formation and
accretion and Active Galactic Nuclei genesis.

The freedom of choice of coordinates in Relativity means that intuition about physical effects becomes complicated. Poor coordinate choices may appear innocuous, but can have severe effects on the long-term evolution of a system. A substantial aspect of the course will be the study of different coordinate choices. Additionally, formulations of General Relativity must be carefully considered for their behavior under perturbation (i.e. the hyperbolicity of the system, which determines how the solutions react to inevitable small computational errors).

At the same time as the discussion of astrophysics proceeds, various
approaches
to discretization and solution of the computational equations (finite
arithmetic, *N*-body approach) will be described.
Actually, modern results in both classical and relativistic regimes are
beyond analytic treatment and require sophisticated numerical
algorithms. Furtherly, realistic astrophysical simulations are so
demanding computationally as to be necessarily implemented on huge
parallel computing platforms. Carrying out such simulations involves a
substantial supercomputing challenge.