Towards physically realizable and hyperbolic moment closures for kinetic theory

Groth, C. P. T.; McDonald, James G.

doi:10.1007/s00161-009-0125-1

Cited by 61 publications

(76 citation statements)

References 37 publications

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“…In transport and kinetic theory, entropy-based closures are used to derive moment models which retain fundamental properties of the underlying kinetic equations such as hyperbolicity, entropy dissipation, and positivity. The resulting models have been studied extensively in the areas of extended thermodynamics [18,48], gas dynamics [24,29,32,33,40,42,57], semiconductors [2][3][4][5]26,31,34,41,56], quantum fluids [16,19], radiative transport [9-12, 20-22, 28, 30, 46, 47, 59, 62], and phonon transport in solids [19]. In spite of their attractive mathematical properties and wide application, entropy methods still suffer from several short-comings.…”

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confidence: 99%

High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

Alldredge

Hauck²,

Tits³

2012

SIAM J. Sci. Comput.

187

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confidence: 99%

High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

Alldredge

Hauck²,

Tits³

2012

SIAM J. Sci. Comput.

187

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“…However, it is also known [17,20,21,38] that the defining optimization problem in this case is ill-posed. As a result, alternative approaches are currently being pursued which regularize the problem in some suitable fashion; see [15] and references therein. For charge transport in semiconductors, the issue of ill-posedness also exists for the so-called parabolic band approximation [29, p.69].…”

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confidence: 99%

High-order entropy-based closures for linear transport in slab geometry

Hauck

2011

Communications in Mathematical Sciences

107

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Abstract. We compute high-order entropy-based (M N ) models for a linear transport equation on a one-dimensional slab geometry. We simulate two test problems from the literature: the twobeam instability and the plane-source problem. In the former case, we compute solutions for systems up to order N = 6; in the latter, up to N = 15. The most notable outcome of these results is the existence of shocks in the steady-state profiles of the two-beam instability for all odd values of N .Key words. Particle transport, maximum entropy, moment closures.AMS subject classifications. 82C70, 82D75, 94A17. IntroductionIn transport and kinetic theory, moment models are used to reduce the size of the state space required for a kinetic description while still maintaining basic features of a kinetic model. They do so by replacing the velocity component of phase space by a finite number of velocity moments. Moment models are commonly derived using an approximate reconstruction of the kinetic description from these moments. The reconstruction prescribes a closure, i.e. a recipe for expressing the moment model as a closed system of the retained moments. Entropy-based methods specify this reconstruction as the solution to a constrained, convex optimization problem. In many situations the cost functional for the optimization problem is directly related to the kinetic entropy of the system. In other cases it simply enforces physically relevant features. The benefit of the entropy approach is a reduced model which retains fundamental properties from the kinetic formalism not found in traditional moment models-properties such as hyperbolicity, entropy dissipation, and positivity. The main disadvantage of the entropy approach is that, unlike traditional moment models, the entropy-based kinetic reconstruction can rarely be expressed as an analytic function of the given moments. Thus the optimization problem must be solved numerically via the associated dual problem. This can increase computational costs significantly.Recent advances in both analysis and implementation of entropy-based methods have been made in several application areas. For gas dynamics, the formal properties of entropy-based models were elucidated in [26]. However, it is also known [17,20,21,38] that the defining optimization problem in this case is ill-posed. As a result, alternative approaches are currently being pursued which regularize the problem in some suitable fashion; see [15] and references therein. For charge transport in semiconductors, the issue of ill-posedness also exists for the so-called parabolic band approximation [29, p.69]. However, for experimental dispersion relations or for more realistic approximations, like the Kane dispersion relation [3, p.3], the optimization

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“…A truncated finite subset of moments, as used in the MOMIC approach, is not sufficient to uniquely reconstruct the PSDF without further assumptions. By checking the realizability conditions, however, one can determine whether there exists at least one "physically plausible, positive semidefinite, distribution function" for the given set of moments (Groth and McDonald 2009).…”

Section: Realizability Of the Soot Psdfmentioning

confidence: 99%

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

Roy

Arias

Lecoustre

et al. 2014

Aerosol Science and Technology

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The method of moments with interpolative closure (MOMIC) for soot formation and growth provides a detailed modeling framework maintaining a good balance in generality, accuracy, robustness, and computational efficiency. This study presents several computational issues in the development and implementation of the MOMIC-based soot modeling for direct numerical simulations (DNS). The issues of concern include a wide dynamic range of numbers, choice of normalization, high effective Schmidt number of soot particles, and realizability of the soot particle size distribution function (PSDF). These problems are not unique to DNS, but they are often exacerbated by the high-order numerical schemes used in DNS. Four specific issues are discussed in this article: the treatment of soot diffusion, choice of interpolation scheme for MOMIC, an approach to deal with strongly oxidizing environments, and realizability of the PSDF. General, robust, and stable approaches are sought to address these issues, minimizing the use of ad hoc treatments such as clipping. The solutions proposed and demonstrated here are being applied to generate new physical insight into complex turbulence-chemistry-soot-radiation interactions in turbulent reacting flows using DNS.

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Towards physically realizable and hyperbolic moment closures for kinetic theory

Cited by 61 publications

References 37 publications

High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

High-order entropy-based closures for linear transport in slab geometry

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

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