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A detailed analytical and numerical analysis of a recently introduced stochastic model for fluid dynamics with continuous velocities and efficient multi-particle collisions is presented. It is shown how full Galilean invariance can be achieved for arbitrary Mach numbers and how other low temperature anomalies can be removed. The relaxation towards thermal equilibrium is investigated numerically, and the relaxation time is measured. Equations of motions for the correlation functions of coarse-grained hydrodynamic variables are derived using a discrete-time projection operator technique, and the Green-Kubo relations for all relevant transport coefficients are given. In the following paper (Part 2), analytic expressions for the transport coefficients are derived and compared with simulation results. Long-time tails in the velocity and stress autocorrelation functions are measured and shown to be in good agreement with previous mode-coupling theories for continuous systems

Stochastic rotation dynamics. I. Formalism, Galilean invariance, and Green-Kubo relations