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

Radtke, Gregg Arthur; Hadjiconstantinou, Nicolas G.; Wagner, Wolfgang

doi:10.1063/1.3562666

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…Such results are significant because they demonstrate that simulations with approximately ten particles per cell are achievable in mass-conservative LVDSMC simulations without apparent random walks in any non-negligible hydrodynamic variables, a substantial improvement over the previous implementation. 26 This dramatic improvement enables efficient simulation in multiple spatial dimensions, as we show below. Here, the time step was chosen as ⌬t = ⌬x / c 0 , where ⌬x is the smallest cell dimension, which effectively treats the effect of ⌬x and ⌬t as a single convergence parameter ͑⌬x͒.…”

Section: Resultsmentioning

confidence: 93%

“…The more general nonlinear approach has been published in a preliminary study. 26 The key in efficiently simulating collisions while maintaining stability in the LVDSMC method lies in exploiting a special representation, 6,12,13,19,20,23 …”

Section: A Collision Stepmentioning

confidence: 99%

“…Finally, while the previous method 21 is strictly valid only for small perturbations from equilibrium, here we generalize the method for all regimes; a version without mass conservation was presented in a previous paper. 26 The advection step simulates the left-hand side of the Boltzmann equation ͓Eq. ͑1͔͒, i.e.,…”

Section: B Advection Stepmentioning

confidence: 99%

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Low-noise Monte Carlo simulation of the variable hard sphere gas

2011

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We present an efficient particle simulation method for the Boltzmann transport equation based on the low-variance deviational simulation Monte Carlo approach to the variable-hard-sphere gas. The proposed method exhibits drastically reduced statistical uncertainty for low-signal problems compared to standard particle methods such as the direct simulation Monte Carlo method. We show that by enforcing mass conservation, accurate simulations can be performed in the transition regime requiring as few as ten particles per cell, enabling efficient simulation of multidimensional problems at arbitrarily small deviation from equilibrium.

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Section: Resultsmentioning

confidence: 93%

Section: A Collision Stepmentioning

confidence: 99%

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Low-noise Monte Carlo simulation of the variable hard sphere gas

2011

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“…On the one hand, this setup provides the background for proving convergence. On the other hand, the tools from Markov process theory helped to generalize the method from hard spheres to the variable hard sphere case, which is often preferred by engineers (see [15] concerning numerical tests). This illustrates that the study of the convergence behaviour of stochastic particle schemes and their relationship with kinetic equations, beside its theoretical interest, can be useful also for practical applications.…”

Section: Relation To Kinetic Equationsmentioning

confidence: 99%

Stochastic models in kinetic theory

Wagner

2011

Physics of Fluids

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The paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (mono-molecular processes), or via binary interactions (bi-molecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent.

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“…Unfortunately, VR DSMC has a numerical instability, similar to the other variance-reduced methods for dilute gases [3,4]; fortunately, in the VR DSMC method this can be overcome [1] using kernel density estimation. The other alternative is to use the stable LVDSMC method of Homolle and Hadjiconstantinou [5,6], which has recently been extended to the VHS model in [7].…”

Section: Introductionmentioning

confidence: 99%

Variance-Reduced Direct Simulation Monte Carlo with the Bhatnagar-Gross-Krook Collision Operator

Landon¹,

Hadjiconstantinou²

2011

AIP Conference Proceedings

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Direct Simulation Monte Carlo (DSMC) is perhaps the most prevalent stochastic method for simulating rarefied gas flows. High-signal flows (e.g. Mach number greater than 0.1) are efficiently resolved using DSMC; but low-signal flows require drastically increased statistical sampling. To address this limitation, Al-Mohssen and Hadjiconstantinou [1] presented a variance-reduced DSMC algorithm that dramatically improves the signal-to-noise ratio of low-signal flows. This variance reduction is achieved by exploiting a nearby, analytically-known equilibrium using importance weights. The weights are updated according to rules derived from the Boltzmann equation. The Bhatnagar-Gross-Krook (BGK) collision operator is a simplistic approximation of the collision term in the Boltzmann equation that is useful in a number of fields involving particle-mediated transport. In this work, we show that the BGK collision operator lends itself naturally to the application of variance reduction using weights by allowing the derivation of weight-update rules from first principles. This feature removes the instabilities introduced by more complex collision rules and produces a stable variance-reduced particle method. We validate the method by comparing to analytic solutions and numerical results from other BGK particle methods.

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

Cited by 3 publications

References 15 publications

Low-noise Monte Carlo simulation of the variable hard sphere gas

Low-noise Monte Carlo simulation of the variable hard sphere gas

Stochastic models in kinetic theory

Variance-Reduced Direct Simulation Monte Carlo with the Bhatnagar-Gross-Krook Collision Operator

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