A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature-both the origin of the discrete-velocity space and the magnitudes of the discrete velocities are dependent on the local flow-and are used in a finite-volume context. The numerical implementation of the model follows the near-equilibrium flow method of Nadiga and Pullin and results in a scheme which is second order in space (in the smooth regions and between first and second order at discon-
tinuities) and second order in time. (The three-dimensional code is ih~luded.) For one choice of the scaling between the magnitude of the discrete velocities and the local internal energy of the flow, the method reduces to a flux-splitting scheme based on characteristics. As a preliminary exercise, the result of the Sod shock-tube simulation is compared to the exact solution.