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 ͑Ma͒ 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 MaӶ 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 MaӶ 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.

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We present and discuss a variance-reduced stochastic particle

SUMMARYWe present and discuss an efficient, high-order numerical solution method for solving the Boltzmann equation for low-speed dilute gas flows. The method's major ingredient is a new Monte Carlo technique for evaluating the weak form of the collision integral necessary for the discontinuous Galerkin formulation used here. The Monte Carlo technique extends the variance reduction ideas first presented in Baker and Hadjiconstantinou (Phys. Fluids 2005; 17, art. no. 051703) and makes evaluation of the weak form of the collision integral not only tractable but also very efficient. The variance reduction, achieved by evaluating only the deviation from equilibrium, results in very low statistical uncertainty and the ability to capture arbitrarily small deviations from equilibrium (e.g. low-flow speed) at a computational cost that is independent of the magnitude of this deviation. As a result, for low-signal flows the proposed method holds a significant computational advantage compared with traditional particle methods such as direct simulation Monte Carlo (DSMC).

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