Investigating the use of low-discrepancy sequences in particle simulations for rarefied gas flows

Matthew McNenly, University of Michigan

Non-continuum, non-equilibrium gas flows are characterized by an insufficient number of particle collisions occurring in the length scale of interest to drive the flow to local thermodynamic equilibrium (LTE).  Traditional continuum-based solution methods, such as the Navier-Stokes equation, are not physically accurate for gas flows that are not near LTE.  For these non-equilibrium flows, alternative methods, like the full Boltzmann simulation or particle based simulations, are needed to maintain the physical accuracy of the results.  The most popular particle method is direct simulation Monte Carlo (DSMC), which enjoys a relatively straightforward implementation that samples the solution of the Boltzmann equation without explicitly solving the non-linear, integro-differential equation.  While DSMC is successful in accurately simulating a wide range of non-continuum, non-equilibrium flows, it suffers from long computation times in the region of low-speed rarefied flows associated with fluidic micro-electro-mechanical systems (MEMS).  The approximate error of any Monte Carlo method decreases as the square root of the number of samples.  Therefore, each additional digit of accuracy requires 100 times as many samples.  The problem with MEMS flows is that the average velocity of interest is much smaller than the random or thermal velocity of the gas, requiring a tremendous number of samples.  In this study, efforts are made to reduce the number of samples, and thus the simulation time required by the particle simulation, by replacing the random sequences used in the DSMC method with low-discrepancy sequences which have a near-linear error convergence.  A quasi-Monte Carlo method based on the improved convergence of the low-discrepancy sequences is developed for the simulation of free molecular flow to serve as a guideline for future improvements of particle methods.

Abstract Author(s): Matthew J. McNenly and Iain D. Boyd