Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation

Abstract

We present an efficient variance-reduced particle simulation technique for solving the linearized Boltzmann transport equation in the relaxation-time approximation used for phonon, electron, and radiative transport, as well as for kinetic gas flows. The variance reduction is achieved by simulating only the deviation from equilibrium. We show that in the limit of small deviation from equilibrium of interest here, the proposed formulation achieves low relative statistical uncertainty that is also independent of the magnitude of the deviation from equilibrium, in stark contrast to standard particle simulation methods. Our results demonstrate that a space-dependent equilibrium distribution improves the variance reduction achieved, especially in the collision-dominated regime where local equilibrium conditions prevail. We also show that by exploiting the physics of relaxation to equilibrium inherent in the relaxation-time approximation, a very simple collision algorithm with a clear physical interpretation can be formulated.

Low-variance direct Monte Carlo simulations using importance weights

“…f eq = f MB,loc ) results in a substantial reduction in the KDE bias. This is in qualitative agreement with previous work [19,23], which reports performance improvements when a variable equilibrium distribution is used.…”

“…For example, the present formulation can be readily extended to other molecular-interaction models [2]; in contrast, extension of LVDSMC to other molecular models -other than the relaxation-time approximation [23] -is more challenging [28]. Another situation where the present formulation holds an advantage is more complex collision processes, such as chemically reacting flows.…”

“…The LVDSMC method has been recently theoretically analyzed by Wagner [28], who also proposed algorithms for simulating the Variable Hard-Sphere (VHS) model [7]. Moreover, LVDSMC has been extended to treat the relaxation-time approximation by Radtke and Hadjiconstantinou [23].…”

Deviational methods for small-scale phonon transport

“…Although a spatially variable control leads to a moderately more complex algorithm, it provides additional freedom for tailoring the control to the local conditions and thus improving variance reduction. As has been shown in previous work [49,51], the computational benefit from this approach can be significant [51], particularly in the limit Kn → 0 where the distribution function is described very well [39] by a spatially dependent equilibrium distribution (known as the local equilibrium). In fact, this endows methods employing variable controls with the ability to efficiently simulate multiscale phenomena, since in regions where kinetic effects are not important (Kn → 0), very few particles are used, and the system is primarily described by the deterministic control.…”

“…One manifestation of this property is that, in contrast to standard particle simulation methods which become increasingly more expensive as the NSF (Kn → 0) limit is approached (larger length scales imply not only more simulation particles, but also longer evolution timescales), deviational methods with a spatially variable equilibrium distribution become more efficient as this limit is approached, because they are able to relegate increasingly more of the description to the equilibrium part, thus reducing the number of particles required for the simulation [49]. Methods with variable equilibrium distributions have been developed for dilute gases [2,51] and shown to exhibit improved variance reduction, particularly as Kn → 0, as expected. In these formulations, the variable control was implemented as piecewise constant within each cell, requiring source terms resulting from the discontinuities in the control at cell boundaries and making multidimensional implementations complex [49,51].…”