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We present a new 3D general relativistic hydrodynamics code for simulationsof stellar core collapse to a neutron star, as well as pulsations andinstabilities of rotating relativistic stars. It uses spectral methods forsolving the metric equations, assuming the conformal flatness approximation forthe three-metric. The matter equations are solved by high-resolutionshock-capturing schemes. We demonstrate that the combination of a finitedifference grid and a spectral grid can be successfully accomplished. This"Mariage des Maillages" (French for grid wedding) approach results in highaccuracy of the metric solver and allows for fully 3D applications usingcomputationally affordable resources, and ensures long term numerical stabilityof the evolution. We compare our new approach to two other, finite differencebased, methods to solve the metric equations. A variety of tests in 2D and 3Dis presented, involving highly perturbed neutron star spacetimes and(axisymmetric) stellar core collapse, demonstrating the ability to handlespacetimes with and without symmetries in strong gravity. These tests are alsoemployed to assess gravitational waveform extraction, which is based on thequadrupole formula.Comment: 29 pages, 16 figures; added more information about convergence tests  and grid setu

Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations