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

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2. Flux-limiting and two-moment transport

The essential simplification in both flux-limited diffusion and the two-moment approach consists in discarding the detailed angular information contained in the radiation field. Instead one considers angular averages (`moments') of the distribution function. The first three are the radiative energy density E, energy flux F , and pressure tensor P , respectively. The basic assumption is that these quantities suffice as a physical description of the radiation field. The moments must satisfy the energy and momentum equations

[EQUATION]

[EQUATION]

which are obtained by angular integrations of the radiative transport equation. Here [FORMULA] and [FORMULA] are the absorptive and total transport opacities, respectively, and [FORMULA] is the `blackbody' thermal energy density. In these equations, the velocity of light has been put equal to one. Eqs. (1) and (2) may be read as monochromatic (spectral) as well as energy integrated equations.

The classical closure problem is that, because there are more physical variables than equations, an additional relation must be supplied to close the set. One important closure is the diffusion approximation (Fick's Law)

[EQUATION]

where [FORMULA], the ratio of mean free path to energy scale height, is the Knudsen number. Where small opacities or steep gradients make [FORMULA], Fick's Law would allow an acausal flux, [FORMULA]. The flux-limiting remedy is to modify (3) to

[EQUATION]

where the flux limiter [FORMULA] is specifically designed to meet the causality requirement

[EQUATION]

and the correct diffusion limit

[EQUATION]

Flux-limited diffusion, therefore, is a minimal moment approach, taking into account only the energy Eq. (1) as an angular moment equation, with a closure at the lowest level.

In the two-moment description closure is expressed by two Eddington factors

[EQUATION]

It is usually assumed that p does not explicitly depend on the energy density, i.e., a `one-dimensional' closure prescription, [FORMULA], is adopted. In principle, however, one has a `two-dimensional' relationship

[EQUATION]

among the first three reduced moments

[EQUATION]

of the radiative distribution function [FORMULA]. (Here [FORMULA] is the direction of the momentum vector of the radiation quanta.) In systems with local axial symmetry (such as plane and spherical geometries), Eq. (8) reduces to a scalar relationship, [FORMULA], because there is a preferred direction. The variable Eddington factor [FORMULA] must satisfy

[EQUATION]

in order that the radiation field have the correct diffusive and free-streaming limits. The constraint (Levermore 1984)

[EQUATION]

follows from f and p being normalized averages of a distribution, cf. Eq. (9). Note that the quantities (9) are spectral and space & time dependent through [FORMULA].

As Eqs. (4) and (7) show, a flux limiter [FORMULA] is directly related to the Eddington flux factor f. Any variable Eddington factor p may be used to construct a flux limiter according to (Levermore 1984)

[EQUATION]

where the inequality follows from (11). Levermore (1984) presents a variety of closures with their associated flux limiters.

Two-moment transport (TMT) and flux-limited diffusion (FLD) are thus closely related. In TMT the closure is at the level of the momentum equation (`2nd moment closure'), while in FLD the momentum equation is formally ignored (`1st moment closure'). It is possible to quantify the error in the momentum balance (Cernohorsky & van den Horn 1990) and to compensate for it by introducing an `artificial' opacity (Janka 1991) into the Knudsen parameter R, Eq. (3), which effectively reinstates the neglected momentum contributions. In this way a modified FLD scheme results which should correctly account for energy-momentum balance. Such a scheme was first presented in the context of neutrino transport by Janka (1991) and Dgani & Janka (1992) as an alternative approach to TMT with a variable Eddington factor. However, while the conceptual framework of FLD is formally preserved, the extended FLD scheme is equivalent to the set of TMT equations. Strictly then, one is no longer solving a diffusion equation, i.e., a parabolic partial differential equation. The moment equations are actually a hyperbolic set.

The hyperbolicity of the moment equations has implications for any approximate solution procedure involving a closure on the variable Eddington factor. In particular, with a nonlinear closure, physically acceptable solutions meeting prescribed boundary conditions may be out of (numerical) reach. As shown by Körner & Janka (1992), the solutions contain a critical point, so that nearby solutions easily diverge away from the physical solution. Smit et al. (1997) have shown under what conditions the physical solution is stably and accurately mimicked.

A hyperbolic system admits discontinuity waves. In the free-streaming case, singularities must progagate with the speed of light. This causality requirement implies (Anile et al. 1991)

[EQUATION]

This constraint on the closure is supplementary to the set (10)-(11), but has not been imposed in standard closures. In the next section, we will see that (13) is met by all fermionic maximum entropy closures, but not by the Wilson and Levermore-Pomraning closures.

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000
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