Commun. Comput. Phys., 15 (2014), pp. 797-826. |
Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers Houde Han ^{1}, Min Tang ^{2*}, Wenjun Ying ^{2} 1 Department of Mathematical Sciences, Tsinghua University, Haidian, Beijing 100084, P.R. China.2 Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P.R. China. Received 13 April 2013; Accepted (in revised version) 1 August 2013 Available online 3 December 2013 doi:10.4208/cicp.130413.010813a 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 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, both methods can not only capture the diffusion limit, but also exhibit uniform convergence 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, 35Q70Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: Neutron transport equation, discrete ordinates method, tailored finite point method, boundary layers, interface layers. *Corresponding author. Email: hhan@math.tsinghua.edu.cn (H. Han), tangmin@sjtu.edu.cn (M. Tang), wying@sjtu.edu.cn (W. Ying) |