We have investigated the use of DSMC as a pseudo-Euler solver in the continuum limit by using a modification of Pullin's Equilibrium Particle Simulation Method (EPSM). EPSM is a particle-based method which is in effect the large collision rate limit of DSMC yet requires far less computational effort. We propose a modification of EPSM, the Particle Flux Method (PFM), which is intermediate between EPSM and a conventional finite volume continuum flow solver. The total mass, momentum and energy in each cell are stored. Flux particles are created at every time step and move in free flight over a short decoupling time step, carrying mass momentum and energy between cells. The new method has been demonstrated by calculating the hypersonic flow over a wedge, for which DSMC calculations are available. Because of an inherent dissipation, related to the cell size and time step, the shock was thicker than that found in the DSMC calculations, but the shock location was the same. PFM is not prohibitively expensive and may have some advantages over conventional continuum based flow solvers, in terms of robustness arising from its firm basis in the physics of molecular flow.