Alternative to the Pomraning-Eddington approach to radiative transfer

The modified Eddington approximation, introduced in 1969 by Pomraning [J. Quant. Spectrosc. Radiat. Transfer 9, 407 (1969)], is used for solving the radiative-transfer equation for plane-parallel media. The specific intensity I(x,μ) is expressed as the sum of an even and an odd function of the angular variable μ, i.e., I(x,μ)=E(x)ε(x,μ)+F(x)o(x,μ); the integro-differential transport equation is thereby transformed into two coupled first-order differential equations involving the energy density E(x), the radiative flux F(x), and D(x)≡${\mathcal{F}}_{\mathrm{-}1}^1$ ${\mathrm{\mu }}^2$ε(x,μ)dμ. In Pomraning’s treatment, one of these two exact equations is replaced by an approximate version derived by neglecting the spatial dependence of D(x); despite this simplification, the coupled equations usually defy, if one is treating inhomogeneous media, attempts at analytic solutions; the boundary conditions for the two differential equations are fabricated by multiplying the condition satisfied by I(x,μ) at each boundary by a prescribed weight function and integrating with respect to μ. We also outline an alternative strategy wherein the integral equations satisfied by D(x), E(x), and F(x) are deduced and solved by means of the variational method. The advantages of the latter approach are twofold. First, since the pertinent boundary conditions are automatically incorporated in the integral equations, the problem of inferring suitable weight functions is obviated; second, the spatial dependence of D(x) is taken into account explicitly. Numerical results are presented to illustrate the performance of the adapted Pomraning-Eddington approach.