The Bird algorithm is a computationally efficient method for simulating dilute gas flows. However, due to the relatively large transport coefficients at low densities, high Rayleigh or Reynolds numbers are difficult to achieve by this technique. We present a modified version of the Bird algorithm in which the relaxation processes are enhanced and the transport coefficients reduced, while preserving the correct equilibrium and nonequilibrium fluid properties. The present algorithm is found to be two to three orders of magnitude faster than molecular dynamics for simulating complex hydrodynamical flows.