A Practical Variance Reduced DSMC Method

Abstract

The goal of this work is to develop a practical method for simulating low-signal kinetic (and small-scale) gaseous flows. These flows have recently received renewed attention in connection with the design, manufacturing, and optimization of MEMS/NEMS devices operating in gaseous environments; they are typically described using the Boltzmann equation which is most efficiently solved using the direct simulation Monte Carlo (DSMC) method. DSMC is a simple and versatile simulation method which is very efficient in producing samples of the single particle distribution function used for estimating hydrodynamic properties. Unfortunately, in the case of low-speed flows the computational cost associated with reducing the statistical uncertainty of simulation outputs becomes overwhelming. We will present a variance reduction approach for reducing the statistical uncertainty associated with low-signal flows making their simulation not only possible but also efficient. Variance reduction is achieved using a control variate approach based on the observation that low-signal flows are typically close to an equilibrium state. As with previous variance reduction methods, significant variance reduction is achieved making the simulation of arbitrarily small deviations from equilibrium possible. However, in contrast to previous variancereduction methods, the method proposed, which we will refer to as the VRDSMC method, is able to reduce the variance with virtually no 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. By retaining the basic DSMC formulation, in contrast to previous approaches, the VRDSMC approach combines ease of implementation with computational efficiency and the ability to simulate all molecular interaction models available within the DSMC formulation. The work presented here represents a substantial improvement from the work presented in the previous symposium in two important ways. First, the kernel density estimation stabilization scheme has been further refined to allow substantially less bias without dramatically affecting stability. The second major improvement is the use of local cell reference states when performing particle collisions which results in a substantial reduction in the number of particles per cell required for stability, especially for small Knudsen numbers. Our validation tests show that the proposed VRDSMC method provides considerable variance reduction for only a small increase in computational cost and approximation error compared to equivalent DSMC simulations. In other words, by addressing the major weakness associated with DSMC, VRDSMC is well suited to the solution of low-signal kinetic problems of practical interest.

Low Variance Particle Simulations of the Boltzmann Transport Equation for the Variable Hard Sphere Collision Model

“…More recently, the same approach was applied to the linearized BGK model [11], which does not require particle cancellation. A weight-based approach for variance reduction was recently introduced by Al-Mohssen and Hadjiconstantinou [12]; this formulation, known as VRDSMC, achieves variance reduction with minimal changes to the original DSMC algorithm.…”