  We investigate the dynamics and evolution of coalescing neutron stars.
Although the code (Piecewise Parabolic Method) is purely Newtonian, we do
include the emission of gravitational waves and their backreaction on the
hydrodynamic flow. The properties of neutron star matter are described by the
physical equation of state of Lattimer \& Swesty (1991). Energy loss by all
types of neutrinos and changes of the electron fraction due to the emission of
electron neutrinos and antineutrinos are taken into account by an elaborate
``neutrino leakage scheme''. We simulate the coalescence of two identical, cool
neutron stars with a baryonic mass of $\approx\!1.6\,M_\odot$ and a radius of
$\approx\!15$~km and with an initial center-to-center distance of 42~km. The
initial distributions of density and electron concentration are given from a
model of a cold neutron star in hydrostatic equilibrium (central temperature
about $8\,{\rm MeV}$). We investigate three cases which differ by the initial
velocity distribution in the neutron stars, representing different cases of the
neutron star spins relative to the direction of the orbital angular momentum
vector. Within about 1~ms the neutron stars merge into a rapidly spinning
($P_{\rm spin}\approx 1$~ms), high-density body ($\rho\approx
10^{14}$~g/cm$^3$) with a surrounding thick disk of material with densities
$\rho\approx 10^{10}-10^{12}$~g/cm$^3$ and orbital velocities of~0.3--0.5~c. In
this work we evaluate the models in detail with respect to the gravitational
wave emission using the quadrupole approximation. In a forthcoming paper we
will concentrate on the neutrino emission and implications for gamma-ray
bursters. A maximum luminosity in excess of $10^{55}$~erg/s is reached for
about 1~ms.
