  Numerical relativity simulations are essential to study the last stages of
the binary neutron star coalescence. Unfortunately, for stable simulations
there is the need to add an artificial low-density atmosphere. Here we discuss
a new framework in which we can effectively set the density surrounding the
neutron stars to zero to ensure a more accurate simulation. We test our method
with a number of single star test cases and for an equal mass binary neutron
star simulation. While the bulk motion of the system is not influenced, and
hence, there is no improvement with respect to the emitted gravitational-wave
signal, we find that the new approach is superior with respect to mass
conservation and it allows a much better tracking of outward moving material.
This will allow a more accurate simulation of the ejected material and supports
the interpretation of present and future multi-messenger observations with more
accurate numerical relativity simulations.
