A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface
Shi Jin, Min Ming Tang, Houde Han · Networks and Heterogeneous Media · 2009
In this paper, we propose a uniformly second order numerical method for the discete-ordinate transportequation in the slab geometry in the diffusive regimes withinterfaces. At the interfaces, the scattering coefficients havediscontinuities, so suitable interface conditions are needed todefine the unique solution. We first approximate the scatteringcoefficients by piecewise constants determined by their cellaverages, and then obtain the analytic solution at each cell, usingwhich to piece together the numerical solution with the neighboringcells by the interface conditions. We show that this method isasymptotic-preserving, which preserves the discrete diffusion limitwith the correct interface condition. Moreover, we show that ourmethod is quadratically convergent uniformly in the diffusiveregime, even with the boundary layers. This is 1) the first sharp uniform convergence result for linear transport equations inthe diffusive regime, a problem that involves both transport anddiffusive scales; and 2) the first uniform convergence valid upto the boundary even if the boundary layers exist, so the boundarylayer does not need to be resolved numerically. Numerical examplesare presented to justify the uniform convergence.