  We present a code for solving the coupled Einstein-hydrodynamics equations to
evolve relativistic, self-gravitating fluids. The Einstein field equations are
solved in generalized harmonic coordinates on one grid using pseudospectral
methods, while the fluids are evolved on another grid using shock-capturing
finite difference or finite volume techniques. We show that the code accurately
evolves equilibrium stars and accretion flows. Then we simulate an equal-mass
nonspinning black hole-neutron star binary, evolving through the final four
orbits of inspiral, through the merger, to the final stationary black hole. The
gravitational waveform can be reliably extracted from the simulation.
