Entropic derivation of the spectral Eddington factors

Larecki, Wieslaw; Banach, Zbigniew

doi:10.1016/j.jqsrt.2011.06.011

Cited by 12 publications

(38 citation statements)

References 46 publications

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“…Remark 1. Following [27], in Definition 1, and in the rest of this paper, we exclude the cases f = 0 and f = 1 almost everywhere (a.e.) on S 2 , which would give J = 0, H = 0 and J = 1, H = 0, respectively.…”

Section: Moment Realizability For the Fermionic Two-moment Modelmentioning

confidence: 99%

“…The ME closure constructs an approximation of the angular distribution as a function of J and H [3,27]. The ME distribution f ME is found by maximizing the entropy functional, which for particles obeying Fermi-Dirac statistics is given by…”

Section: Maximum Entropy (Me) Closurementioning

confidence: 99%

“…More recently, Larecki & Banach [27] have shown that the explicit expression given in [3] is not exact and provide another approximate expression…”

Section: Maximum Entropy (Me) Closurementioning

confidence: 99%

“…red line in Figure 10). We have also run this test using the algebraic maximum entropy closure of Larecki & Banach [27] and the simpler Kershaw-type closure in [4]. The numerical solutions obtained with both of these closures follow the analytic solution well, and remain within the realizable set R. We point out that simply using the realizabilitypreserving limiter described in Section 8 with the Minerbo closure does not result in a realizability-preserving scheme for Fermi-Dirac statistics because of the properties of this closure discussed in Section 4, and plotted in the right panel of Figure 3.…”

Section: Packed Beammentioning

confidence: 99%

“…In the fermionic case, there is also an upper bound on the distribution function (e.g., f ≤ 1) because Pauli's exclusion principle prevents particles from occupying the same microscopic state. The fermionic two-moment model has recently been studied theoretically in the context of maximum entropy closures [27,28,29] and Kershaw-type closures [4]. Because of the upper bound on the distribution function, the algebraic constraints on realizable moments differ from the classical case with no upper bound, and can lead to significantly different dynamics when the occupancy is high (i.e., when f is close to its upper bound).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Chu

Endeve

Hauck

et al. 2019

Journal of Computational Physics

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Building on the framework of Zhang & Shu [1,2], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function f . The two-moment model is closed using algebraic moment closures; e.g., as proposed by Cernohorsky & Bludman [3] and Banach & Larecki [4]. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., f ∈ [0, 1]). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. The IMEX scheme is formally only first-order accurate, but works well in the diffusion limit, and -without interactions with the background -reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of Shu & Osher [5]. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport. (Ran Chu), endevee@ornl.gov (Eirik Endeve), hauckc@ornl.gov (Cory D. Hauck), mezz@utk.edu (Anthony Mezzacappa) both spectral and finite volume methods and are an attractive option for solving hyperbolic partial differential equations (PDEs). They achieve high-order accuracy on a compact stencil; i.e., data is only communicated with nearest neighbors, regardless of the formal order of accuracy, which can lead to a high computation to communication ratio, and favorable parallel scalability on heterogeneous architectures has been demonstrated [32]. Furthermore, they can easily be applied to problems involving curvilinear coordinates (e.g., beneficial in numerical relativity [33]). Importantly, DG methods exhibit favorable properties when collisions with a background are included, as they recover the correct asymptotic behavior in the diffusion limit, characterized by frequent collisions (e.g., [34,35,36]). The DG method was introduced in the 1970s by Reed & Hill [37] to solve the neutron transport equation, and has undergone remarkable developments since then (see, e.g., [38] and references therein).We are concerned with t...

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Section: Moment Realizability For the Fermionic Two-moment Modelmentioning

confidence: 99%

Section: Maximum Entropy (Me) Closurementioning

confidence: 99%

“…More recently, Larecki & Banach [27] have shown that the explicit expression given in [3] is not exact and provide another approximate expression…”

Section: Maximum Entropy (Me) Closurementioning

confidence: 99%

Section: Packed Beammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Chu

Endeve

Hauck

et al. 2019

Journal of Computational Physics

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Kershaw-type transport equations for fermionic radiation

Banach

Larecki

2017

Z. Angew. Math. Phys.

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Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae

Mezzacappa

Endeve

Messer

2020

Living Rev Comput Astrophys

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The proposal that core collapse supernovae are neutrino driven is still the subject of active investigation more than 50 years after the seminal paper by Colgate and White. The modern version of this paradigm, which we owe to Wilson, proposes that the supernova shock wave is powered by neutrino heating, mediated by the absorption of electron-flavor neutrinos and antineutrinos emanating from the proto-neutron star surface, or neutrinosphere. Neutrino weak interactions with the stellar core fluid, the theory of which is still evolving, are flavor and energy dependent. The associated neutrino mean free paths extend over many orders of magnitude and are never always small relative to the stellar core radius. Thus, neutrinos are never always fluid like. Instead, a kinetic description of them in terms of distribution functions that determine the number density of neutrinos in the six-dimensional phase space of position, direction, and energy, for both neutrinos and antineutrinos of each flavor, or in terms of angular moments of these neutrino distributions that instead provide neutrino number densities in the four-dimensional phase-space subspace of position and energy, is needed. In turn, the computational challenge is twofold: (i) to map the kinetic equations governing the evolution of these distributions or moments onto discrete representations that are stable, accurate, and, perhaps most important, respect physical laws such as conservation of lepton number and energy and the Fermi–Dirac nature of neutrinos and (ii) to develop efficient, supercomputer-architecture-aware solution methods for the resultant nonlinear algebraic equations. In this review, we present the current state of the art in attempts to meet this challenge.

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Entropic derivation of the spectral Eddington factors

Cited by 12 publications

References 46 publications

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Kershaw-type transport equations for fermionic radiation

Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae

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