Error Analysis of Variations on Larsen’s Benchmark Problem

Error norms for three variants of Larsen’s benchmark problem are evaluated using three numerical methods for solving the discrete ordinates approximation of the neutron transport equation in multidimensional Cartesian geometry. The three variants ,of Larsen’s test problem are concerned with the incoming flux boundary conditions: unit incoming flux on the left and bottom edges (Larsen’s configuration); unit, incoming flux only on the left edge; unit incoming flux only on the bottom edge. The three methods considered are the Diamond Difference (DD) method, and the constant-approximation versions of the Arbitrarily High Order Transport method of the Nodal type (AHOT-N), and of the Characteristic (AHOT-C) type. The cell-wise error is computed as the difference between the cell-averaged flux computed by each method and the exact value, then the Lr, La, and L, error norms are calculated. The results of this study demonstrate that while integral error norms, i.e. L1, Lz, converge to zero with mesh refinement, the pointwise L, norm does not due to solution discontinuity across the singular characteristic. Little difference is observed between the error norm behavior of the three methods considered in spite of the fact that AHOT-C is locally exact, suggesting that numerical diffusion across the singular characteristic as the major source of error on the global scale. However, AHOT-C possesses a given accuracy in a larger fraction of computational cells than DD.