Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Trans- port Equations with Boundary and Interface Layers

Abstract. This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime, 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 in-troduced. The schemes firstly 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, both methods can not only capture the diffusion limit, but also exhibit uniform conver-gence in the diffusive regime, even with boundary layers. Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy, uniformly with respect to the mean free path. Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers. AMS subject classifications: 65L12, 76N20, 35Q70 Key words: Neutron transport equation, discrete ordinates method, tailored finite point method