Acceleration of multidimensional discrete ordinates methods via adjacent-cell preconditioners

The adjacent-cell preconditioner (AP) formalism originally derived in slab geometry is extended to multidimensional Cartesian geometry for genetic fix ed-weight, weighted diamond difference neutron transport methods, This is a ccomplished for the thick-cell regime (KAP) and thin-cell regime (NAP). A s pectral analysis of the resulting acceleration schemes demonstrates their e xcellent spectral properties for model problem configurations, characterize d bf a uniform mesh of infinite extent and homogeneous material composition each in its own cell-size regime. Thus, the spectral radius of KAP vanishe s as the computational cell size approaches infinity but it exceeds unity f or very thin cells, thereby implying instability In contrast, NAP is stable and robust for all cell sizes, but its spectral radius vanishes more slowl y as the cell size increases. For this reason, and to avoid potential compl ication in the case of cells that are thin in one dimension and thick in an other, NAP is adopted in the remainder of this work. The most important fea ture of AP for practical implementation in production level codes is that i t is cell centered, reducing the size of the algebraic system comprising th e acceleration stage compared to face-centered schemes. Boundary conditions for finite extent problems and a mixing formula across material and cell-s ize discontinuity are derived and used to implement NAP in a test rode, AHO T, and a production code, TORT. Numerical testing for algebraically linear iterative schemes for the cases embodied in Burre's Suite of Test Problems demonstrates the high efficiency of the new method in reducing the number o f iterations required to achieve convergence, especially for optically thic k cells where acceleration is most needed. Also, for algebraically nonlinea r (adaptive) methods, AP generally performs better than the par tial curren t rebalance method in TORT and the diffusion synthetic acceleration method in TWODANT. Finally, application of the AP formalism to a simplified linear nodal (SLN) method similar, but not identical, to TORT's linear nodal opti on is shown to possess two eigenvalues that approach either one or infinity with increasing cell size regardless of the preconditioner parameters. Thi s implies impossibility of unconditionally robust acceleration of SLN-type methods with cell-centered preconditioners that have a block-diffusion coup ling stencil Edge-centered acceleration methods, or methods that do not req uire the linear moments of the flux to converge, might have an advantage in this regard but at a significant penalty to computational efficiency due t o the larger system soh,ed or the inability to utilize the computed linear moments.