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Astron. Astrophys. 356, 559-569 (2000)

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1. Introduction

In radiative systems radiation is invariably transported through the medium. The transport equation, therefore, is the basic equation underlying the radiative hydrodynamics. In practice one commonly circumvents this fundamental equation by resorting to some simplifying procedure to approximate the transport problem. Although computationally intensive, the Boltzmann equation can nowadays be solved numerically in one spatial dimension. Nevertheless, semi-analytic approximation schemes are needed in higher dimensional problems, and are often more illuminating than exact treatments, even in one dimension. The physical foundation for a particular choice of approximation, however, is not always in evidence, and the ensuing physical description of the system may be rather qualitative.

Currently fashionable approximations are flux-limiting prescriptions and closure relations for the moment equations, a number of which we shall examine below. Although flux-limiting and the moment method are essentially different approaches to deal with anisotropic radiation fields, the two are connected through a generic relationship between flux limiter and Eddington factor (cf Sect. 2). In this paper we focus on two-moment transport, and check a number of closure relations against direct numerical solutions of the Boltzmann equation. We will do so within the scope of fermionic (neutrino) radiation. Previous investigations in this field (Janka 1991, 1992; Janka et al. 1992; Cernohorsky & Bludman 1994) have revealed a number of shortcomings of standard closures and have proposed possible improvements. One such improved treatment is provided by two-dimensional maximum entropy closure. While the numerical overhead of any two-dimensional closure would easily appear at odds with the attempt for computational economy, it has proven possible, in the case of Fermi-Dirac statistics, to formulate an efficient closure algorithm (Cernohorsky & Bludman 1994).

There is no such thing as the "correct" closure. At most one may strive for a closure which is able to describe the radiation field "as well as possible" in a given transport problem. The quick way is to adopt an ad hoc relation, for example one that smoothly interpolates between the diffusive and free-streaming fluxes, such as Wilson's closure (Sect. 3.2.1). Or one may look for such a relation based on geometrical or other considerations. Alternatively, the closure can be derived from a given or assumed angular dependence of the radiative distribution function. In some cases the functional dependence obtained from direct transport calculations may serve to model the closure, as is the case for Janka's Monte Carlo closure (Sect. 3.2.3).

An appealing approach is to derive the angular dependence from a basic principle. In this spirit the maximum entropy closure (Minerbo 1978) and the Levermore-Pomraning closure (Levermore & Pomraning 1981), discussed in Sects. 3.1 and 3.2.2, have been obtained. These were derived originally for the case of photon radiation, and have subsequently been applied to neutrino transport as well. For photon radiation more closures can be found in the literature (Levermore 1984).

The Wilson and Levermore-Pomraning closures have become standard in neutrino transport calculations. Implementation is commonly effected through use of the corresponding flux limiters in a flux-limited diffusion (FLD) scheme. In our calculations for this paper we employed a two-moment transport (TMT) scheme incorporating maximum entropy closure (MEC) besides various other closures. The philosophy behind MEC is that it allows for the least biased distribution of the radiation quanta based on the available information, viz. particle statistics, energy or occupation density, and flux. The dependence on energy density in addition to the flux calls for an inherently two-dimensional closure which may contain more traditional one-dimensional closures as limiting cases.

If TMT with a given closure is to be successful, the TMT solution should approximate, as closely as possible, the first three angular moments of the exact distribution, for each point in space and for every energy. Beyond this "weak equivalence" of the angular moments, one may also consider "strong equivalence" of a given model distribution, i.e., judging whether or not the exact distribution is well represented by the model distribution (Cernohorsky & Bludman 1994).

In Sect. 2 we briefly outline the procedures of flux-limited diffusion and two-moment transport. Though offering different perspectives, an intimate relation exists between their central concepts of flux limiter and closure. Various closures are reviewed in Sect. 3. In Sect. 4 the validity of approximate transport is evaluated in terms of the concepts of weak and strong equivalence of the angular distribution and its moments. For this purpose we consider two different material backgrounds and a number of neutrino energies. We also examine the behavior of spectral and of energy averaged Eddington factors. Among the closures considered, two-dimensional MEC appears to give the best overall approximation to the full Boltzmann transport calculations, while the Wilson and Levermore-Pomraning closures are poorest. Our conclusions are summarized in Sect. 5.

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000