Subcell balance methods for radiative transfer on arbitrary grids

Abstract

We present a new spatial discretization method. which enforces conservation on quadrilateral subcells in an arbitrarily connected grid of polygonal cells, for two-dimensional radiative transfer problems. We review what is known about the performance of existing methods for optically thick, diffusive regions of radiative transfer problems, focusing in particular on bilinear discontinuous (BLD) finitezlement methods and the simple cornerhalance (SCB) method. We discuss the close relation of the SCB and BLD methods, and how they differ. By careful analysis, we relate specific properties of the SCB solution to specific approximations in the SCB method. We then build our new method by discarding those SCB approximations that lead to undesirable properties and carefully constructing new approximations designed to yield more desirable properties. We compare BLD, SCB. and the new scheme on a series of test problems in slab and XY geometries; numerical results invariably agree with predictions of the analysis. The new method matches SCB in thick diffusive regions. as it was designed to, but vastly outperforms SCB given cells of fine and intermediate optical thickness, where it achieves results comparable to BLD.

Radiation Hydrodynamics

“…The problems associated with the unhappy choice between inaccuracy (step differencing) and negative solutions (typified by the diamond-difference method) have vanished today, though the use of the discontinuous finite-element method (cf., [85]) and the new corner-balance method of refs. [62,2].…”

“…The thermalization depth is about equal to the net distance the photon will go in that many flights, which as we have seen is about e ¤ N. Therefore the thermalization depth is approximately 12½ on the ¤ scale for the case of Doppler broadening. This does vary with the shape of the line profile, so for example, for a Lorentzian profile the thermalization depth is roughly 12½ 2 . A more precise definition of the thermalization depth ù that is consistent with this argument is given by the statement…”

“…In its simplest incarnation, for 1-D slab geometry, it is described as follows. The material of the problem is divided into N zones with fixed masses, divided by The expansions agree through terms of order 2 , and so the numerical method is second-order accurate. The third-order terms differ except in the case C 1.…”

Diffusion Limit of the Lorentz Model: Asymptotic Preserving Schemes

“…There is a large number of applications based on transport equations having a diffusive asymptotic. Let us mention, for instance, neutron transport, radiative transfer in the "optically thick limit" (see [1,31,32,38,39] and the references therein) and semiconductor modeling (see [25,29,37]). However, many of the previous works deal with the so-called telegrapher equation, or equivalently Goldstein-Taylor equation which is a kinetic equation where the distribution function is localized on two opposite velocities (see [4,20,21,26,27,40]).…”

Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods

“…Their method is very much like the Galerkin formulations that are used in computational #uid dynamics (CFD) and even uses some concepts of upwinding to achieve second-order accuracy. Adams [20] and Adams and Novak [21] have designed and analyzed a simple corner balance (SCB) method that is very much like the lumped linear-discontinuous scheme in certain limits. Even parity, "nite-element method (EP-FEM)-based solution techniques for the radiative transfer equation were explored in Fiveland and Jessee [22,23].…”