An exponential discontinuous scheme for X-Y-Z geometry transport problems Page: 2 of 9
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AN EXPONENTIAL DISCONTINUOUS SCHEME FOR
X-Y-Z GEOMETRY TRANSPORT PROBLEMS
Todd A. Wareing and Raymond E. Alcouffe
Los Alamos National Laboratory
P.O. Box 1663
Los Alamos, NM 87545
The recently developed exponential discontinuous spatial differencing scheme for the discrete-ordinate
equations has been extended to x-y-z geometry with hexahedral cells. This scheme produces strictly
positive angular fluxes given positive discrete-ordinate sources. The exponential discontinuous scheme
has been developed and implemented into the three-dimensional, discrete-ordinate code, THREEDANT.
Numerical results are given which show that the exponential discontinuous scheme is very accurate for
deep-penetration transport problems with optically thick spatial meshes.
Recently, the algebraically equivalent nonlinear characteristic1-3 (NC) and exponential characteristic4-5
(EC) spatial differencing schemes were developed. These schemes use the method of characteristics with
an exponential representation for the discrete-ordinate source within each spatial cell. This source
representation is derived using information theory6. The NC/EC scheme is very accurate, especially for
deep-penetration problems with optically thick spatial cells. The NC/EC scheme also produces strictly
positive angular fluxes given positive discrete-ordinate sources. Recently we developed an exponential
discontinuous (ED) scheme in x-y geometry based upon an exponential representation for the angular flux
within each cell7. This representation is also derived using information theory. The motivation for the ED
scheme is that it is much less complicated to derive and implement than the NC/EC scheme and is assumed
to be considerably less computationally expensive in x-y-z geometry. The NC/EC scheme is exact for
pure-absorbing problems in all Cartesian geometries. The ED scheme is only exact for pure-absorbing
problems in slab geometry. Nonetheless, the ED scheme appears to be nearly as accurate as the NC/EC
scheme for many problems and also produces strictly positive angular fluxes. The ED scheme has been
extended to x-y-z geometry with a hexahedral mesh and has been implemented into the three-dimensional,
discrete-ordinate code, THREEDANTTM 8.
The remainder of the paper will proceed as follows: in Sec. II, we derive the ED scheme in x-y-z
geometry and discuss the method of solution; in Sec. III, we present some numerical results to show the
accuracy and positivity of the ED scheme; and in Sec IV, we conclude with a discussion.
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Wareing, T.A. & Alcouffe, R.E. An exponential discontinuous scheme for X-Y-Z geometry transport problems, report, April 1, 1996; New Mexico. (digital.library.unt.edu/ark:/67531/metadc670768/m1/2/: accessed January 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.