## 1. IntroductionIn 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 ( 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 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, 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, 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. © European Southern Observatory (ESO) 2000 Online publication: April 10, 2000 |