Multilevel asymptotic-preserving Monte Carlo for kinetic-diffusive particle simulations of the Boltzmann-BGK equation

Mortier, Bert; Robbe, Pieterjan; Baelmans, Martine; Samaey, Giovanni

doi:10.1016/j.jcp.2021.110736

Cited by 4 publications

(10 citation statements)

References 25 publications

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“…To analyze the significance these effects, we intend to perform a more efficient implementation in a compiled language in future work. Another potential area for future work, is to apply the approach in Section 3.2 to the scheme presented in [35] and compare both schemes.…”

Section: Discussionmentioning

confidence: 99%

“…The weak diffusive process produced by (3.2) does not have the same distribution as the Brownian process used in the independent simulation of (2.7)-(2.8) at level 0, unless M(v) ≡ N (0, 1). A similar issue arises in the alternate scheme in [35]. For large M , one can assume that the law of large numbers will result in a suitably small bias.…”

Section: If There Is No Collision In Fine Timementioning

confidence: 95%

“…We also perform a detailed numerical analysis of the new algorithm. We also refer to [35], where similar ideas were used to develop an alternate APMLMC scheme based on the AP scheme in [34].…”

mentioning

confidence: 99%

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Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlo

Løvbak¹,

Samaey²

2022

Preprint

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Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a Quantity of Interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in highcollisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research.

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Section: Discussionmentioning

confidence: 99%

Section: If There Is No Collision In Fine Timementioning

confidence: 95%

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Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlo

Løvbak¹,

Samaey²

2022

Preprint

Self Cite

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show abstract

“…[6] One class of APMC methods is the kinetic-diffusion algorithms. [2] These exhibit superlinear convergence [13] and bounded simulation cost towards the kinetic limit. We focus specifically on the KDMC algorithm for which a multi-level extension has been developed.…”

Section: An Apmc Method: Kdmcmentioning

confidence: 99%

“…These methods are unbiased but can require prohibitive amounts of computational time. Next, the APMC method [2,13] is presented that is much more efficient at the expense of an asymptotically vanishing bias. In Section 3, the new estimation method is presented and in Section 4, the new ideas are applied in three incremental numerical experiments on a fusion-relevant test-case.…”

Section: Introductionmentioning

confidence: 99%

Estimation as a post‐processing step for random walk approximations of the Boltzmann‐BGK model

Mortier

Maes

Samaey

2022

Contributions to Plasma Physics

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Recently, asymptotic‐preserving Monte Carlo simulation methods have been developed to simulate the Boltzmann‐BGK equation with advective–diffusive limiting behaviour over a broad range of regimes. These simulation methods hybridize a particle tracing Monte Carlo scheme for the kinetic equation and a random walk (RW) Monte Carlo simulation for the advection–diffusion limit, combining the precision of the former with the efficiency of the latter. In the RW part of the simulation, details of the travelled path are absent. This complicates the extraction of integral quantities of interest such as mass, momentum, and energy sources. Here, we present a new estimation strategy that couples the RW parts of the particle trajectories with a deterministic simulation of the corresponding fluid model. The contributions of the RW parts to quantities of interest are then computed from a single time step of this fluid model in a post‐processing step. We illustrate this new estimation strategy for a fusion‐relevant test‐case and focus on the bias and variance present when using this estimator.

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Fluid, kinetic and hybrid approaches for neutral and trace ion edge transport modelling in fusion devices

et al. 2022

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Neutral gas physics and neutral interactions with the plasma are key aspects of edge plasma and divertor physics in a fusion reactor including the detachment phenomenon often seen as key to dealing with the power exhaust challenges. A full physics description of the neutral gas dynamics requires a 6D kinetic approach, potentially time dependent, where the details of the wall geometry play a substantial role, to the extent that, e.g., the subdivertor region has to be included. The Monte Carlo (MC) approach used for about 30 years in EIRENE [1], is well suited to solve these types of complex problems. Indeed, the MC approach allows simulating the 6D kinetic equation without having to store the velocity distribution on a 6D grid, at the cost of introducing statistical noise. MC also provides very good flexibility in terms of geometry and atomic and molecular (A&M) processes. However, it becomes computationally extremely demanding in high-collisional regions (HCR) as anticipated in ITER and DEMO. Parallelization on particles helps reducing the simulation wall clock time, but to provide speed-up in situations where single trajectories potentially involve a very large number of A&M events, it is important to derive a hierarchy of models in terms of accuracy and to clearly identify for what type of physics issues they provide reliable answers. It was demonstrated that advanced fluid neutral (AFN) models are very accurate in HCRs, and at least an order of magnitude faster than fully kinetic simulations. Based on these fluid models, three hybrid fluid-kinetic approaches are introduced: a spatially hybrid technique (SpH), a micro-Macro hybrid method (mMH), and an asymptotic-preserving MC (APMC) scheme, to combine the efficiency of a fluid model with the accuracy of a kinetic description. In addition, atomic and molecular ions involved in the edge plasma chemistry can also be treated kinetically within the MC solver, opening the way for further hybridisation by enabling kinetic impurity ion transport calculations. This paper aims to give an overview of methods mentioned and suggests the most prospective combinations to be developed.

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Multilevel asymptotic-preserving Monte Carlo for kinetic-diffusive particle simulations of the Boltzmann-BGK equation

Cited by 4 publications

References 25 publications

Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlo

Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlo

Estimation as a post‐processing step for random walk approximations of the Boltzmann‐BGK model

Fluid, kinetic and hybrid approaches for neutral and trace ion edge transport modelling in fusion devices

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