By means of Burgers' equation it is shown that a recently proposed stochastic formulation of fluid dynamics allows for a new approach to numerical studies of hydrodynamic fluctuations. The new formulation describes the velocity field as the expectation value of a discrete stochastic process governed by a multivariate master equation. With the help of a systematic expansion it is demonstrated that the fluctuations of this discrete stochastic process correspond to those predicted by the theory of fluctuating hydrodynamics. Stochastic simulations of shock waves are presented and stress the practical relevance of the new technique.