Direct simulation Monte Carlo convergence behavior of the hard-sphere-gas thermal conductivity for Fourier heat flow

The convergence behavior of the direct simulation Monte Carlo (DSMC) method is systematically investigated for near-continuum, one-dimensional Fourier flow. An argon-like, hard-sphere gas is confined between two parallel, fully accommodating, motionless walls of unequal temperature. The simulations are performed using four variations based on Bird's DSMC algorithm that differ in the ordering of the move, collide, and sample operations. The primary convergence metric studied is the ratio of the DSMC-calculated bulk thermal conductivity to the infinite-approximation Chapman-Enskog (CE) theoretical value, although temperature and heat flux are also considered. Ensemble, temporal, and spatial averaging are used to reduce statistical errors to levels that are small compared to the discretization errors from the time step (Delta t), the cell size (Delta x), and the number of computational particles per cell (N-c). The errors from these three parameters are determined using over 700 individual cases selected from the ranges 0.05 <= 480. The infinite-particle-number (N-c ->infinity) convergence behavior for the thermal-conductivity ratio is found to be second-order in both time step and cell size, in good agreement with previous theoretical predictions based on Green-Kubo theory. For vanishing time step and cell size, the finite-particle-number convergence behavior is found to be O(1/N-c) if similar to 30 or more particles per cell are used. The observed convergence behavior is found to be more complicated when all three discretization parameters are finite. As discretization errors are systematically reduced, the DSMC-calculated conductivity is shown to approach the infinite-approximation CE theoretical value to within 1 part in 10(4).