A formulation of hydrodynamics in terms of discrete cell variables is given, which is as exact as possible in the sense that, for given initial measurements on a system, it gives the maximum possible information on future measurements. An apparently rapidly convergent successive-approximation scheme is described, which requires data obtainable from short-time computer simulation on small systems. The lowest approximation gives the linearized Navier-Stokes and energy flow equations, yielding explicit expressions for transport coefficients.