## 2. Flux-limiting and two-moment transportThe 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 which are obtained by angular integrations of the radiative transport equation. Here and are the absorptive and total transport opacities, respectively, and 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) where , the ratio of mean free path to energy scale height, is the Knudsen number. Where small opacities or steep gradients make , Fick's Law would allow an acausal flux, . The flux-limiting remedy is to modify (3) to where the and the correct diffusion limit 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 It is usually assumed that p does not explicitly depend on the
energy density, among the first three reduced moments of the radiative distribution function . (Here 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, , because there is a preferred direction. The variable Eddington factor must satisfy in order that the radiation field have the correct diffusive and free-streaming limits. The constraint (Levermore 1984) follows from As Eqs. (4) and (7) show, a flux limiter
is directly related to the Eddington
flux factor 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 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) 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. © European Southern Observatory (ESO) 2000 Online publication: April 10, 2000 |