A Linear-Discontinuous Spatial Differencing Scheme forSnRadiative Transfer Calculations
✍ Scribed by J.E. Morel; Todd A. Wareing; Kenneth Smith
- Book ID
- 102585956
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 452 KB
- Volume
- 128
- Category
- Article
- ISSN
- 0021-9991
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✦ Synopsis
the diamond difference scheme [1], can yield large and negative discrete solutions in optically thick cells and are Various types of linear-discontinuous spatial differencing schemes have been developed for the S n (discrete-ordinates) equa-thus unacceptable for radiative transfer calculations. There tions approximating the linear Boltzmann transport equation. It has are other simple schemes, such as the step or upwind been shown through an asymptotic analysis that the 1D slab-geomescheme [1], which are strictly positive. However, the step try lumped linear-discontinuous scheme not only goes over to a scheme can give very poor results in diffusive regions when convergent and robust differencing of the diffusion equation in the the cells are optically thick. Counter to intuition, this can be monoenergetic thick diffusion limit, but it also yields the correct interior solution, even when boundary layers are left unresolved the case even when the analytic solution varies arbitrarily by the spatial mesh. In the present work we generalize this scheme slowly over a cell. The theoretical justification for this beto obtain a 1D slab-geometry lumped linear-discontinuous scheme havior arises from the fact that truncation error analysis for the nonlinear radiative transfer equation and the associated indicates that consistent spatial discretization schemes for material temperature equation. We present a full nonlinear energydependent asymptotic analysis of the behavior of this scheme in the S n equations necessarily converge only in the limit as the thick equilibrium-diffusion limit. We find that this scheme goes the cell width becomes small relative to a mean-free-path.
over to a convergent and robust differencing of the equilibrium-A form of analysis known as a thick diffusion-limit analysis diffusion equation on the interior of the mesh, but it does not yield has been developed by Larsen et. al. [2] to provide informathe exact interior solution when boundary layers are left unresolved tion on the behavior of S n spatial discretization schemes by the spatial mesh. Nevertheless, the interior solution obtained with spatially unresolved boundary layers is always well behaved when the analytic solution is diffusive and the spatial cells and fairly accurate. Computational results are presented which test are optically thick. The idea of this analysis is fairly straightthe predictions of our asymptotic analysis and demonstrate the forward and can be described as follows. The diffusion efficiency of our solution technique.