Resolution of singularities of the diamond difference approximation

Journal Article · Mon Dec 01 04:00:00 EST 1980 · SIAM J. Numer. Anal.; (United States)

DOI:https://doi.org/10.1137/0717067· OSTI ID:6697510

Abu-Shumays, I K

The standard diamond difference approximation to the transport theory discrete ordinates equations in x-y geometry leads to singular systems of equations for cell (reflecting) and for some periodic boundary conditions. As a consequence: (a) the solutions of the matrix approximations, though unique at centers of mesh cells, are physically meaningless at cell boundaries, and (b) in general, convergence of applicable iterative methods is rather slow in comparison to the case when vacuum boundary conditions are used. To overcome these difficulties, a slightly less accurate (first order) weighted diamond difference approximation is generally applied. Often numerical schemes which set negative fluxes to zero and readjust the computations are also applied, though at an increase in the computional cost. The presence of negative flux solutions generally indicates that either the problem formulation or the mesh subdivision is inadequate. This paper shows how a proper reinterpretation of the physical boundary conditions can remove the singularity of the diamond difference discretization. It is proved that a discrete L/sub 2/-norm of the resulting numerical approximation to the transport equation has the same order of accuracy as the standard diamond difference approximation.

• Research Organization: Bettis Atomic Power Lab., West Mifflin, PA
• OSTI ID: 6697510
• Journal Information: SIAM J. Numer. Anal.; (United States), Journal Name: SIAM J. Numer. Anal.; (United States) Vol. 17:6; ISSN SJNAA
• Country of Publication: United States
• Language: English

Related Subjects

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73 NUCLEAR PHYSICS AND RADIATION PHYSICS
DISCRETE ORDINATE METHOD
MATHEMATICAL MODELS
NEUTRON TRANSPORT THEORY
NUMERICAL SOLUTION
TRANSPORT THEORY