Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers (Communications in Computational Physics, 2014)
This paper presents two uniformly convergent numerical schemes for thetwo dimensional steady state discrete ordinates transport equation in the diffusiveregime, which is valid up to the boundary and interface layers. A five-point node-centered and a four-point cell-centered tailored finite point schemes (TFPS) are introduced. The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system. Numerically, bothmethods can not only capture the diffusion limit, but also exhibit uniform convergencein the diffusive regime, even with boundary layers. Numerical results show that thefive-point scheme has first-order accuracy and the four-point scheme has second-orderaccuracy, uniformly with respect to the mean free path. Therefore, a relatively coarsegrid can be used to capture the two dimensional boundary and interface layers.