Dynamic Fisheye Grids for Binary Black Holes Simulations
  We present a new warped gridding scheme adapted to simulating gas dynamics in
binary black hole spacetimes. The grid concentrates grid points in the vicinity
of each black hole to resolve the smaller scale structures there, and rarefies
grid points away from each black hole to keep the overall problem size at a
practical level. In this respect, our system can be thought of as a "double"
version of the fisheye coordinate system, used before in numerical relativity
codes for evolving binary black holes. The gridding scheme is constructed as a
mapping between a uniform coordinate system--in which the equations of motion
are solved--to the distorted system representing the spatial locations of our
grid points. Since we are motivated to eventually use this system for
circumbinary disc calculations, we demonstrate how the distorted system can be
constructed to asymptote to the typical spherical polar coordinate system,
amenable to efficiently simulating orbiting gas flows about central objects
with little numerical diffusion. We discuss its implementation in the Harm3d
code, tailored to evolve the magnetohydrodynamics equations in curved
spacetimes. We evaluate the performance of the system's implementation in
Harm3d with a series of tests, such as the advected magnetic field loop test,
magnetized Bondi accretion, and evolutions of hydrodynamic discs about a single
black hole and about a binary black hole. Like we have done with Harm3d, this
gridding scheme can be implemented in other unigrid codes as a (possibly)
simpler alternative to adaptive mesh refinement.
