On the quasi-high-order property of weighted linear discontinuous methods

Szilard and Pomraning introduced the one-parameter (θ) family of weighted linear discontinuous spatial approximations to the monoenergetic discrete ordinates equations in plane-parallel geometry. They concentrated on nonnegative values of θ, observed that Larsen and Morel had suggested that the method corresponding to θ = 1 might be advantageous for meshes having optically thick cells (i.e., it satisfies an asymptotic diffusive limit) and noted that θ → ∞ yields the classical diamond-difference method. As regards accuracy of the approximate cell-edge angular fluxes in the fine-mesh limit, Szilard and Pomraning showed that all methods with nonnegative θ have a truncation error of order at least two, except that the method corresponding to θ = 3 (the classical linear discontinuous method) has order of at least three, while that corresponding to θ = 0 (the step differencing scheme) displays order of at least one. Their computational results for the most part confirmed that these estimates hold exactly; however, they noted the anomalous result for some situations, particularly for computation of reflection and absorption in cases with small transmission (i.e., approximately a half-space) and no internal sources, that third-order accuracy seems to prevail over a wide range of values of θ. The purpose of the work summarized here was to investigate further this anomalous case