Diffusion Synthetic Acceleration Methods for the Diamond-Differenced Discrete-Ordinates Equations

Abstract

We investigate a class of acceleration schemes that resemble the conventional synthetic method in that they utillze the difiusion operator in the transport iteration schemes. The acctilerated iteration involves alternate dif~usion and transport solutj.onswhere coupling between the equations is achieved using a correction term applied to either (1) the diffusion coefficient, (2) the removal cross-section; or (3) the source of the diffusion equation. Tl,emethods involving the modification of the diffusion coefficient and of the removal term yield nonlinear acceleration schemes and are used in k calculations, eff while the source term modification approach is linear at-least before discretization, and used fur inhomogr.neous source problems. there is n preferred diffcrencing method observed instability of the preferred difference scheme at the same time stable and conventional A careful analysis shows that which eliminates the previously synthetic method. Using this results in m acceleration method which is efficient. This preferred difference approach renders the source correction scheme, which is linear in Its continuous * Work performed under the auspices of the U. S. Energy Research and Development Administration.

Splitting d'opérateur pour l'équation de transport neutronique en géométrie bidimensionnelle plane

“…Une accélération de la convergence de cet algorithme dite accélération par diffusion synthétique (DSA) aété introduite par Alcooffe en 1977 [6] dans le cas d'une géométrie 1D plane et aétéétudiée aussi bien théoriquement que numériquement dans le cas de conditions aux limites de type réflexion (i.e. f (x, Ω) = f (x, −Ω), ∀x ∈ ∂D).…”

“…f (x, Ω) = f (x, −Ω), ∀x ∈ ∂D). En 1D cetteétude peutêtre consultée dans [6,9] et en 2D on peut se référerà [1]. Toutefois, dans le cas de conditions aux limites de type flux entrant donné, nous ne disposons d'aucun résultat théorique de convergence de la DSA (voir [12,13]).…”

Multigroup diffusion preconditioners for multiplying fixed-source transport problems

“…Although the consistency of the diffusion discretization used for DSA-preconditioning is not a focus of this paper, we note that the mesh-centered discretization is not consistent with the diamond-difference spatial discretization used for the transport equation for which an edge-centered discretization is known to be consistent [27].…”

Black Box Multigrid with coarsening by a factor of three

“…However, the choice may be dictated by the application. For example, the vertex-centered FVM is necessary for diffusionsynthetic acceleration of the neutron transport equation, as the cell-centered FVM is unstable in this case [17].…”