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Abstract Interpreting gravitational wave observations and understanding the physics of astrophysical compact objects such as black holes or neutron stars requires accurate theoretical models. Here, we present a new numerical relativity computer program, called Nmesh , that has the design goal to become a next generation program for the simulation of challenging relativistic astrophysics problems such as binary black hole or neutron star mergers. In order to efficiently run on large supercomputers, Nmesh uses a discontinuous Galerkin method together with a domain decomposition and mesh refinement that parallelizes and scales well. In this work, we discuss the various numerical methods we use. We also present results of test problems such as the evolution of scalar waves, single black holes and neutron stars, as well as shock tubes. In addition, we introduce a new positivity limiter that allows us to stably evolve single neutron stars without an additional artificial atmosphere, or other more traditional limiters.