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

M. Alcubierre·S. Brandt·B. Bruegmann·D. Holz·E. Seidel·R. Takahashi·J. Thornburg
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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.

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M. Alcubierre, S. Brandt, B. Bruegmann, D. Holz, E. Seidel, R. Takahashi, J. Thornburg. 1999-08-04. Symmetry without Symmetry: Numerical Simulation of Axisymmetric Systems using Cartesian Grids. https://doi.org/10.1142/s0218271801000834

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