Symmetry without symmetry: Numerical simulation of axisymmetric systems using Cartesian grids

Miguel Alcubierre, Bernd Brügmann, Daniel Holz, Ryoji Takahashi, Steven Brandt, Edward Seidel, Jonathan Thornburg

Research output: Contribution to journalArticlepeer-review


We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3 + 1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a three-dimensional Cartesian (x, y, z) coordinate grid which covers (say) the y = 0 plane, but is only one finite-difference-molecule-width thick in the y direction. The field variables in the central y = 0 grid plane can be updated using normal (x, y, z)-coordinate finite differencing, while those in the y ≠ 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and 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.

Original languageEnglish (US)
Pages (from-to)273-289
Number of pages17
JournalInternational Journal of Modern Physics D
Issue number3
StatePublished - Jun 2001
Externally publishedYes

ASJC Scopus subject areas

  • Mathematical Physics
  • Astronomy and Astrophysics
  • Space and Planetary Science


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