Lumping Techniques for DFEM S_N Transport in Slab Geometry

We examine several mass matrix lumping techniques for the discrete ordinates (S_N) particle transport equations spatially discretized with arbitrary order discontinuous finite elements in one-dimensional (1-D) slab geometry. Though positive outflow angular flux is guaranteed with traditional mass matrix lumping for linear solution representations in source-free, purely absorbing 1-D slab geometry, we show that when used with higher-degree polynomial trial spaces, traditional lumping does not yield strictly positive outflows and does not increase the solution accuracy with increase in the polynomial degree of the trial space. As an alternative, we examine quadrature-based lumping strategies, which we term “self-lumping” (SL). Self-lumping creates diagonal mass matrices by using a numerical quadrature restricted to the Lagrange interpolatory points. When choosing equally spaced interpolatory points, SL is achieved through the use of closed Newton-Cotes formulas, resulting in strictly positive outflows for odd degree polynomial trial spaces in 1-D slab geometry. When selecting the interpolatory points to be the abscissas of a Gauss-Legendre or a Lobatto-Gauss-Legendre quadrature, it is possible to obtain solution representations with a strictly positive outflow in source-free pure absorber problems for any degree polynomial trial space in 1-D slab geometry. Furthermore, there is no inherent limit to local truncation error order of accuracy when using interpolatory points that correspond to Gauss-Legendre or Lobatto-Gauss-Legendre quadrature points. A single-cell analysis is performed to investigate outflow positivity and truncation error as a function of the trial space polynomial degree, the choice of interpolatory points, and the numerical integration strategy. We also verify that the single-cell local truncation error analysis translates into the expected global spatial convergence rates in multiple-cell problems.