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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.
@article{M2AN_2010__44_5_1069_0,
author = {Al-Mohssen, Husain A. and Hadjiconstantinou, Nicolas G.},
title = {Low-variance direct {Monte} {Carlo} simulations using importance weights},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1069--1083},
publisher = {EDP-Sciences},
volume = {44},
number = {5},
year = {2010},
doi = {10.1051/m2an/2010052},
mrnumber = {2731403},
zbl = {1200.82051},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010052/}
}
TY - JOUR AU - Al-Mohssen, Husain A. AU - Hadjiconstantinou, Nicolas G. TI - Low-variance direct Monte Carlo simulations using importance weights JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2010 SP - 1069 EP - 1083 VL - 44 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010052/ DO - 10.1051/m2an/2010052 LA - en ID - M2AN_2010__44_5_1069_0 ER -
%0 Journal Article %A Al-Mohssen, Husain A. %A Hadjiconstantinou, Nicolas G. %T Low-variance direct Monte Carlo simulations using importance weights %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2010 %P 1069-1083 %V 44 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010052/ %R 10.1051/m2an/2010052 %G en %F M2AN_2010__44_5_1069_0
Al-Mohssen, Husain A.; Hadjiconstantinou, Nicolas G. Low-variance direct Monte Carlo simulations using importance weights. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 1069-1083. doi: 10.1051/m2an/2010052
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