  We present the first neutron star merger simulations performed with the newly
developed Numerical Relativity code SPHINCS_BSSN. This code evolves the
spacetime on a mesh using the BSSN formulation, but matter is evolved via
Lagrangian particles according to a high-accuracy version of
general-relativistic Smooth Particle Hydrodynamics (SPH). Our code contains a
number of new methodological elements compared to other Numerical Relativity
codes. The main focus here is on the new elements that were introduced to model
neutron star mergers. These include a) a refinement (fixed in time) of the
spacetime-mesh, b) corresponding changes in the particle--mesh mapping
algorithm and c) a novel way to construct SPH initial data for binary systems
via the recently developed "Artificial Pressure Method." This latter method
makes use of the spectral initial data produced by the library LORENE, and is
implemented in a new code called SPHINCS_ID. While our main focus is on
introducing these new methodological elements and documenting the current
status of SPHINCS_BSSN, we also show as a first application a set of neutron
star merger simulations employing "soft" ($\Gamma=2.00$) and "stiff"
($\Gamma=2.75$) polytropic equations of state.
