We present a new methodology for simulating self-gravitating general-relativistic fluids. In our approach the fluid is modelled by means of Lagrangian particles in the framework of a general-relativistic (GR) Smooth Particle Hydrodynamics (SPH) formulation, while the spacetime is evolved on a mesh according to the BSSN formulation that is also frequently used in Eulerian GR-hydrodynamics. To the best of our knowledge this is the first Lagrangian fully general relativistic hydrodynamics code (all previous SPH approaches used approximations to GR-gravity). A core ingredient of our particle-mesh approach is the coupling between the gas (represented by particles) and the spacetime (represented by a mesh) for which we have developed a set of sophisticated interpolation tools that are inspired by other particle-mesh approaches, in particular by vortex-particle methods. One advantage of splitting the methodology between matter and spacetime is that it gives us more freedom in choosing the resolution, so that -- if the spacetime is smooth enough -- we obtain good results already with a moderate number of grid cells and can focus the computational effort on the simulation of the matter. Further advantages of our approach are the ease with which ejecta can be tracked and the fact that the neutron star surface remains well-behaved and does not need any particular treatment. In the hydrodynamics part of the code we use a number of techniques that are new to SPH, such as reconstruction, slope limiting and steering dissipation by monitoring entropy conservation. We describe here in detail the employed numerical methods and demonstrate the code performance in a number of benchmark problems ranging from shock tube tests, over Cowling approximations to the fully dynamical evolution of neutron stars in self-consistently evolved spacetimes.
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