Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes

Larsen, Edward W.; Morel, Jim E.; Miller, W.F.

doi:10.1016/0021-9991(87)90170-7

Cited by 243 publications

(171 citation statements)

References 17 publications

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“…This result can be extended to essentially all AP schemes, although the specific proof is problem dependent. We refer to AP schemes for kinetic equations in the fluid dynamic or diffusive regimes [2,7,14,32,[40][41][42]44,45,[47][48][49]. The AP framework has also been extended in [15,16] for the study of the quasi-neutral limit of Euler-Poisson and Vlasov-Poisson systems, and in [19,21,34] for all-speed (Mach number) fluid equations bridging the passage from compressible flows to the incompressible flows.…”

Section: ð1:4þmentioning

confidence: 99%

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Filbet

Jin

2010

Journal of Computational Physics

320

432

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a b s t r a c tIn this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.

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Section: ð1:4þmentioning

confidence: 99%

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Filbet

Jin

2010

Journal of Computational Physics

320

432

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“…In addition to the difficulty of numerical stiffness arising due to the small scaling, improper underresolved numerical solution often fail to capture the hydrodynamic drift-diffusion limit. Earlier study on numerical methods for transport or kinetic equations indicates that, in order for the underresolved numerical approximation to capture the correct diffusive behavior, the scheme should be asymptotic preserving (AP), in the sense that the asymptotic limit that leads from the transport or kinetic equations to the diffusion equations should be preserved at the discrete level [Ada,Jin,JL1,JL2,JPT1,JPT2,Kl1,Kl2,LMM,LM,Mil,NP1,NP2].…”

Section: Introductionmentioning

confidence: 99%

Discretization of the Multiscale Semiconductor Boltzmann Equation by Diffusive Relaxation Schemes

Jin

Pareschi

2000

Journal of Computational Physics

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“…In the context of linear neutron transport, Larsen and co-workers (Larsen, Morel andMiller 1987, Borgers, Larsen andAdams 1992) have investigated the diffusion limit for a variety of difference schemes and have found that many of them have the correct diffusion limit only in special regimes. They have constructed some alternative methods that always have the correct diffusion limit.…”

Section: Methods With the Correct Diffusion Limitmentioning

confidence: 99%

Monte Carlo and Quasi-Monte Carlo Methods

Jakob¹,

Marschner²

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N~1^2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O ((log N^N' 1 ).For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

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Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes

Cited by 243 publications

References 17 publications

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Discretization of the Multiscale Semiconductor Boltzmann Equation by Diffusive Relaxation Schemes

Monte Carlo and Quasi-Monte Carlo Methods

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