Transport methods: general. 6. A Flux-Limited Diffusion Theory Derived from the Maximum Entropy Eddington Factor

The Minerbo's maximum entropy Eddington factor (MEEF) method was proposed as a low-order approximation to transport theory, in which the first two moment equations are closed for the scalar flux f and the current F through a statistically derived nonlinear Eddington factor f. This closure has the ability to handle various degrees of anisotropy of angular flux and is well justified both numerically and theoretically. Thus, a lot of efforts have been made to use this approximation in transport computations, especially in the radiative transfer and astrophysics communities. However, the method suffers numerical instability and may lead to anomalous solutions if the equations are solved by certain commonly used (implicit) mesh schemes. Studies on numerical stability in one-dimensional cases show that the MEEF equations can be solved satisfactorily by an implicit scheme (of treating δΦ/δx) if the angular flux is not too anisotropic so that f < 0.556 is always satisfied; otherwise, only explicit upwind schemes such as Riemann solvers can produce regular solutions, as demonstrated recently. Therefore, the applicability of the MEEF method is severely limited, because explicit schemes are generally inefficient and not practical for time-independent problems (due to extremely high computational cost), and implicit schemes cannot be used for many problems that involve strong anisotropy in angular flux. To have an implicit method that still utilizes the MEEF closure and is applicable to any problem, one needs to make some approximation to remove the cause of instability in the MEEF description. A flux-limited diffusion theory can be derived from the description by assuming that some combination of the variations of f in space and time is small. This new flux-limited diffusion (NFLD) description to a transport equation with an isotropic source is formally in the same format as the classic diffusion equation. The NFLD method has advantages over the classic diffusion theory: It is more accurate and can handle various problems with strong anisotropy. It is also more convenient to use than the MEEF method especially for time-independent problems. It is proved to be numerically stable and can be solved easily by iterating the diffusion coefficient. This new method has been tested in time-dependent radiative transfer problems and showed good performance. We present here results of two steady-state transport problems to demonstrate its merits. The first problem is a homogeneous slab of 10 mean free paths (mfp's) and scattering ration c = 0.1, with isotropic flux impinging on both sides. The second one is a three-region problem. The middle region of 20 mfp has c = 0.9 and a constant source, and the flanking regions, each of 20 mfp, have c = 0.1 and vacuum boundaries. Displayed in the Figs. 1 and 2 are the transport solution S32, the classic diffusion solution P1, the MEEF solution fM obtained by Riemann solvers, and the NFLD solution DM for the two problems, respectively. In Fig. 1, NFLD and MEEF quantitatively predict very close results. However, the NFLD solution is qualitatively better because it is continuous while MEEF predicts unphysical jumps near the middle of the slab. In Fig. 2, the NFLD and MEEF solutions are almost identical, except near the material interface. In summary, the flux-limited diffusion theory derived from the MEEF description is quantitatively as accurate as the MEEF method. However, it is more qualitatively correct and user-friendly than the MEEF method and can be applied efficiently to various steady-state problems. Numerical tests show that this method is widely valid and overall predicts better results than other low-order approximations for various kinds of problems, including eigenvalue problems. Thus, it is an appealing approximate solution technique that is fast computationally and yet is accurate enough for a wide class of applications