A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface

In this paper, we propose a uniformly second order numerical method for the discete-ordinate transport equation in the slab geometry in the diffusive regimes with interfaces. At the interfaces, the scattering coefficients have discontinuities, so suitable interface conditions are needed to define the unique solution. We first approximate the scattering coefficients by piecewise constants determined by their cell averages, and then obtain the analytic solution at each cell, using which to piece together the numerical solution with the neighboring cells by the interface conditions. We show that this method is asymptotic-preserving, which preserves the discrete diffusion limit with the correct interface condition. Moreover, we show that our method is quadratically convergent uniformly in the diffusive regime, even with the boundary layers. This is 1) the first sharp uniform convergence result for linear transport equations in the diffusive regime, a problem that involves both transport and diffusive scales; and 2) the first uniform convergence valid up to the boundary even if the boundary layers exist, so the boundary layer does not need to be resolved numerically. Numerical examples are presented to justify the uniform convergence.