Variance‐reduced Monte Carlo solutions of the Boltzmann equation for low‐speed gas flows: A discontinuous Galerkin formulation

We 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). Copyright © 2008 John Wiley & Sons, Ltd.