Variance reduction for Monte Carlo solutions of the Boltzmann equation

We show that by considering only the deviation from equilibrium, significant computational savings can be obtained in Monte Carlo evaluations of the Boltzmann collision integral for flow problems in the small Mach number (Ma) limit. The benefits of this variance reduction approach include a significantly reduced statistical uncertainty when the deviation from equilibrium is small, and a flow-velocity signal-to-noise ratio that remains approximately constant with Ma in the Ma⪡1 limit. This results in stochastic Boltzmann solution methods whose computational cost for a given signal-to-noise ratio is essentially independent of Ma for Ma⪡1; our numerical implementation demonstrates this for Mach numbers as low as 10−5. These features are in sharp contrast to current particle-based simulation techniques in which statistical sampling leads to computational cost that is proportional to Ma−2, making calculations at small Ma very expensive.