Error estimates for nodal and short characteristics spatial approximations of two-dimensional discrete ordinates method
Authors: Jose Ignacio Duo
Venue:  (2008)

This work presents advances on error estimation of three spatial approximations of the Discrete Ordinates (DO) method for solving the neutron transport equation. The three methods considered are the Diamond Difference (DD) method, the Arbitrarily High Order Transport method of the Nodal type (AHOT-N), and of the Characteristic type (AHOT-C). The AHOT-N method is employed in constant, linear, quadratic and cubic orders of spatial approximation. The AHOT-C is used in constant, linear and quadratic approximations. Error norms for different problems in non-scattering or isotropic scattering media over two dimensional Cartesian geometries are evaluated. The problems are characterized with different levels of differentiability of the exact solution of the DO equations. The cell-wise error is computed as the difference between the cell-averaged flux calculated by each method and the cell-averaged exact value. The cell error values are used to determine L1, L2, and L&infin; discrete error norms. The results of this analysis demonstrate that while integral error norms, i.e. L1 and L 2, converge to zero with mesh refinement, the cell-wise, L&infin;, norm may not converge when the exact flux possesses discontinuities across the Singular Characteristic (SC). The results suggest that smearing (numerical diffusion) across the SC is the major source of error on the global scale. To mitigate the adverse effect of the smearing, we propose a new Singular Characteristic Tracking (SCT) algorithm which achieves cell-wise convergence even for the cases with discontinuous exact flux. Convergence is restored by hindering numerical diffusion across the SC when resolving the streaming operator in the standard inner sweep iterations. SCT solves two separate Step Characteristics stencils for two sub-cell defined by the intersection of the SC with a mesh cell. Compared to the standard DD, DD-SCT increases the L1 error norm rate of convergence (based on cell size) from 0.5 to 2 for uncollided discontinuous exact flux, and from 0.3 to 1.3 for discontinuous exact flux with isotropic scattering. To provide a confidence level to the spatial resolution of the DO equations, we have casted the AHOT-N method as a Discontinuous Petrov-Galerkin method. Within the mathematical framework of Finite Element Methods (FEM), we have derived an a posteriori error estimator that furnishes a bound on the global L2 error norm. When sufficient regularity is assumed of the adjoint solution, the error estimator is written as a function of the numerical solution&rsquo;s volume and surface residuals, and cell edges discontinuities, for which we present easily computable approximations. As a direct application of decomposing the global error norm estimator into local indicators, we have tested an Adaptive Mesh Refinement (AMR) strategy to enhance computational efficiency without compromising accuracy. In a Shielding Benchmark problem, we show that for the same level of tolerance in the L2 error norm, we can decrease the required number of unknowns (degrees of freedom) by a factor of 10 when comparing AMR to uniform refinement. The ACM Portal is published by the Association for Computing Machinery. Copyright © 2010 ACM, Inc. Terms of Usage Privacy Policy Code of Ethics Contact Us Useful downloads: Adobe Acrobat QuickTime Windows Media Player Real Player