We present results based on an implementation of the Godunov Smoothed Particle Hydrodynamics (GSPH), originally developed by Inutsuka (2002), in the GADGET-3 hydrodynamic code. We first review the derivation of the GSPH discretization of the equations of moment and energy conservation, starting from the convolution of these equations with the interpolating kernel. The two most important aspects of the numerical implementation of these equations are (a) the appearance of fluid velocity and pressure obtained from the solution of the Riemann problem between each pair of particles, and (b the absence of an artificial viscosity term. We carry out three different controlled hydrodynamical three-dimensional tests, namely the Sod shock tube, the development of Kelvin-Helmholtz instabilities in a shear flow test, and the blob test describing the evolution of a cold cloud moving against a hot wind. The results of our tests confirm and extend in a number of aspects those recently obtained by Cha (2010): (i) GSPH provides a much improved description of contact discontinuities, with respect to SPH, thus avoiding the appearance of spurious pressure forces; (ii) GSPH is able to follow the development of gas-dynamical instabilities, such as the Kevin--Helmholtz and the Rayleigh-Taylor ones; (iii) as a result, GSPH describes the development of curl structures in the shear-flow test and the dissolution of the cold cloud in the blob test. We also discuss in detail the effect on the performances of GSPH of changing different aspects of its implementation. The results of our tests demonstrate that GSPH is in fact a highly promising hydrodynamic scheme, also to be coupled to an N-body solver, for astrophysical and cosmological applications. [abridged]
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