The simulation of both low mass and high mass star formation represents a formidable grand challenge in computational astrophysics. Accurate simulation of the huge dynamic range in both the spatial and density scales involved in the presence of strongly turbulent flows requires fully three-dimensional calculations that must self-consistently couple self-gravitational hydrodynamics, radiation transport and magnetic fields. The equations of coupled magneto-radiation-hydrodynamics with self-gravity are solved on a dynamic adaptive mesh refined (AMR) grid on parallel machines allowing a huge dynamic range in scale to be efficiently computed as well as achieving high resolution accuracy in the all the key physics. In this talk I will outline some of the key elements in our multi-physics, parallel AMR code and discuss our recent results in the formation of low mass stars from turbulent cloud cores. I will also present the first self-consistent 3D calculations of the formation of high mass stars showing how the self-consistent turbulent accretion models can overcome the Eddington limit barrier for high mass star formation without the need for competitive accretion. I will present the first results of our newly developed AMR unsplit 3D Godunov MHD code and give rigorous test results for hydrodynamics, radiation and MHD.
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