M. Alcubierre et al., Symmetry without symmetry: Numerical simulation of axisymmetric systems using Cartesian grids, INT J MOD D, 10(3), 2001, pp. 273-289
We present a new technique for the numerical simulation of axisymmetric sys
tems. This technique avoids the coordinate singularities which often arise
when cylindrical or polar-spherical coordinate finite difference grids are
used, particularly in simulating tenser partial differential equations like
those of 3 + 1 numerical relativity. For a system axisymmetric about the r
axis, the basic idea is to use a three-dimensional Cartesian (x,y,z) coord
inate grid which covers (say) the y = 0 plane, but is only one finite-diffe
rence-molecule-width thick in the y direction. The field variables in the c
entral y = 0 grid plane can be updated using normal (x, y, z)-coordinate fi
nite differencing, while those in the y not equal 0 grid planes can be comp
uted from those in the central plane by using the axisymmetry assumption an
d interpolation. We demonstrate the effectiveness of the approach on a set
of fully nonlinear test computations in 3 + 1 numerical general relativity,
involving both black holes and collapsing gravitational waves.