Abstract. We present an efficient approach for reducing the statistical uncertainty associated with direct Monte Carlo simulations of the Boltzmann equation. As with previous variance-reduction approaches, the resulting relative statistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by the characteristic value of quantity of interest) is small and independent of the magnitude of the deviation from equilibrium, making the simulation of arbitrarily small deviations from equilibrium possible. In contrast to previous variance-reduction methods, the method presented here is able to substantially reduce variance with very little modification to the standard DSMC algorithm. This is achieved by introducing an auxiliary equilibrium simulation which, via an importance weight formulation, uses the same particle data as the non-equilibrium (DSMC) calculation; subtracting the equilibrium from the non-equilibrium hydrodynamic fields drastically reduces the statistical uncertainty of the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steady and unsteady, isothermal and non-isothermal problems; in all cases very good agreement between the two methods is found.Mathematics Subject Classification. 60H30, 76P05.

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We present a coarse-grained steady-state solution framework for the Boltzmann kinetic equation based on a Newton-Broyden iteration. This approach is an extension of the equation-free framework proposed by Kevrekidis and coworkers, whose objective is the use of fine-scale simulation tools to directly extract coarse-grained, macroscopic information. Our current objective is the development of efficient simulation tools for modeling complex micro- and nanoscale flows. The iterative method proposed and used here consists of a short Boltzmann transient evolution step and a Newton-Broyden contraction mapping step based on the Boltzmann solution; the latter step only solves for the macroscopic field of interest (e.g., flow velocity). The predicted macroscopic field is then used as an initial condition for the Boltzmann solver for the next iteration. We have validated this approach for isothermal, one-dimensional flows in the low Knudsen number regime. We find that the Newton-Broyden iteration converges in O(10) iterations, starting from arbitrary guess solutions and a Navier-Stokes based initial Jacobian. This results in computational savings compared to time-explicit integration to steady states when the time to steady state is longer than O(40) mean collision times.

We investigate the time evolution of an impulsive start problem for arbitrary Knudsen numbers ($\hbox{\it Kn}$) using a linearized kinetic formulation. The early-time behaviour is described by a solution of the collisionless Boltzmann equation. The same solution can be used to describe the late-time behaviour for $\hbox{\it Kn}\,{\gg}\,1$. The late-time behaviour for $\hbox{\it Kn}\,{<}\,0.5$ is captured by a newly proposed second-order slip model with no adjustable parameters. All theoretical results are verified by direct Monte Carlo solutions of the nonlinear Boltzmann equation. A measure of the timescale to steady state, normalized by the momentum diffusion timescale, shows that the timescale to steady state is significantly extended by ballistic transport, even at low Knudsen numbers where the latter is only important close to the system walls. This effect is captured for $\hbox{\it Kn}\,{<}\,0.5$ by the slip model which predicts the equivalent effective domain size increase (slip length).