High-accuracy numerical simulations of merging neutron stars play an
important role in testing and calibrating the waveform models used by
gravitational wave observatories. Obtaining high-accuracy waveforms at a
reasonable computational cost, however, remains a significant challenge. One
issue is that high-order convergence of the solution requires the use of smooth
evolution variables, while many of the equations of state used to model the
neutron star matter have discontinuities, typically in the first derivative of
the pressure. Spectral formulations of the equation of state have been proposed
as a potential solution to this problem. Here, we report on the numerical
implementation of spectral equations of state in the Spectral Einstein Code. We
show that, in our code, spectral equations of state allow for high-accuracy
simulations at a lower computational cost than commonly used `piecewise
polytrope' equations state. We also demonstrate that not all spectral equations
of state are equally useful: different choices for the low-density part of the
equation of state can significantly impact the cost and accuracy of
simulations. As a result, simulations of neutron star mergers present us with a
trade-off between the cost of simulations and the physical realism of the
chosen equation of state.
