  Errors due to imperfect boundary conditions in numerical relativity
simulations of binary black holes can produce unphysical reflections of
gravitational waves which compromise the accuracy of waveform predictions,
especially for subdominant modes. A system of higher order absorbing boundary
conditions which greatly reduces this problem was introduced in earlier work
[arXiv:gr-qc/0608051]. In this paper, we devise two new implementations of this
boundary condition system in the Spectral Einstein Code (SpEC), and test them
in both linear multipolar gravitational wave and inspiralling mass ratio 7:1
binary black hole simulations. One of our implementations in particular is
shown to be extremely robust and to produce accuracy superior to the standard
freezing-Psi_0 boundary condition usually used by SpEC.
