Symmetry without Symmetry: Numerical Simulation of Axisymmetric Systems using Cartesian Grids

TL;DR

This paper introduces a novel Cartesian grid-based method for simulating axisymmetric systems that avoids coordinate singularities, demonstrated through complex numerical relativity tests involving black holes and gravitational waves.

Contribution

The paper presents a new technique that uses a thin Cartesian grid to simulate axisymmetric systems without coordinate singularities, improving numerical stability and accuracy.

Findings

Effective in nonlinear 3+1 numerical relativity simulations

Handles black holes and gravitational waves accurately

Avoids coordinate singularities in axisymmetric simulations

Abstract

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 3-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 \neq 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.