Several interesting astrophysical phenomena are symmetric with respect to the
rotation axis, like the head-on collision of compact bodies, the collapse
and/or accretion of fields with a large variety of geometries, or some forms of
gravitational waves. Most current numerical relativity codes, however, can not
take advantage of these symmetries due to the fact that singularities in the
adapted coordinates, either at the origin or at the axis of symmetry, rapidly
cause the simulation to crash. Because of this regularity problem it has become
common practice to use full-blown Cartesian three-dimensional codes to simulate
axi-symmetric systems. In this work we follow a recent idea idea of Rinne and
Stewart and present a simple procedure to regularize the equations both in
spherical and axi-symmetric spaces. We explicitly show the regularity of the
evolution equations, describe the corresponding numerical code, and present
several examples clearly showing the regularity of our evolutions.Comment: 11 pages, 9 figures. Several changes. Main corrections are in eqs.
(2.12) and (5.14). Accepted in Gen. Rel. Gra
