The numerical computation of gravitational radiation emitted from binary black hole systems has been pivotal in the detection and characterization of 10 binary black hole astrophysical events since Sept. 14, 2015. However, the drive for more accurate waveforms remains, especially when computing higher-order spherical harmonic modes. The numerical relativity simulations currently used for LIGO / VIRGO detections solve Einstein’s field equations on a finite computational domain with an outer boundary. In order to obtain a unique Cauchy evolution, it is necessary to impose boundary conditions which should yield a well posed problem and ideally, be completely transparent to the physical problem on the unbounded domain. Short of achieving this ideal, one can try to develop so-called absorbing boundary conditions which form a well-posed problem and insure that only a very small amount of spurious gravitational radiation is reflected from the outer boundary into the computational domain. In this talk, I discuss the implementation of a hierarchy of such boundary conditions which hopefully will improve the accuracy of binary black hole numerical simulations.