Towards Improving Binary Black Hole Simulations Using Compact Finite Differences

The accuracy of waveforms from binary black hole simulations will need to increase in the next few years as next generation gravitational wave observatories come online. Current accuracy is sufficient for today's gravitational wave detections. But observatories such as LISA may require an increase in order of accuracy of at least 2 orders of magnitude. Compact Finite Differences (CFDs) have been used increasingly in a variety of engineering applications with a corresponding increase in accuracy. We are working to incorporate compact finite differencing methods in evolution codes for general relativity. Currently, we use Dendro-GR to model large mass ratio binary black holes. These models in particular become more resource heavy as the mass ratio of the binary increases. By using CFDs, we hope to decrease computational times while improving the accuracy. These methods are ultimately implicit schemes but do not dramatically increase computational cost because we can precalculate matrix elements. We have developed a means for calculating these operators based on the order of the derivative, the order of accuracy, and the nature of the banded matrix. An issue that has recently drawn attention is ensuring that the resulting operators are stable. We report on our effort to find stable, highly accurate schemes of first and second derivatives for use in Dendro-GR.