Skip to main content Accessibility help

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

  • Houde Han (a1), Min Tang (a2) and Wenjun Ying (a2)


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.


Corresponding author



Hide All
[1]Adams, M. L., Discontinuous finite element transport solutions in thick diffusive problems, Nucl. Sci. Eng., 137 (2001), 298–333.
[2]Anli, F. and Güngör, S., A spectral nodal method for one-group x,y,z-cartesian geometry discrete ordinates problems, Annals of Nuclear Energy, 23 (1996), 669680.
[3]Azmy, Y. Y., Arbitrarily high order characteristic methods for solving the neutron transport equation, Ann. Nucl. Energy. 19 (1992), 593–606.
[4]Bal, G. and Ryzhik, L., Diffusion Approximation of Radiative Transfer Problems with Interfaces, SIAM, J. Appl. Math., 60(6) (2000),1887–1912.
[5]De, R.C. Barros and Larsen, E.W., A numerical method for one-group slab-geometry discrete ordinates problems with no spatial truncation error, Nuclear Science and Engineering, 104 (1990), 199–208.
[6]Barros, R.C. De and Larsen, E.W., A spectral nodal method for one-group x,y-geometry discrete ordinates problems, Nuclear Science and Engineering, 111 (1992), 34–45.
[7]Brennan, C. R., Miller, R. L., Mathews, K. A., Split-cell exponential characteristic transport method for unstructured tetrahedral meshes, Nucl. Sci. Eng., 138 (2001), 26–44.
[8]Jin, S., Asymptotic preserving (ap) schemes for multiscale kinetic and hyperbolic equations: a review. lecture notes for summer school on “methods and models of kinetic theory” (mmkt), tech. report, Rivista di Mathematica della Universita di Parma, Porto Ercole (Grosseto, Italy), 2010.
[9]Jin, S., Tang, M. and Han, H., A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface, Networks and Heterogeneous Media, 4, (2009), 35–65.
[10]Jin, S., Yang, X. and Yuan, G. W., A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface, Kinetic and Related Models, 1, (2008), 65–84.
[11]Han, H., Miller, J.J.H. and Tang, M., A parameter-uniform tailored finite point method for singularly perturbed linear ODE systems, J. Comp. Math., 31 (2013), 422–438.
[12]Han, H. and Huang, Z., A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium, J. Comp. Math., 26 (2008), 728–739.
[13]Han, H. and Huang, Z., Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions, Journal of Scientific Computing, 41 (2009), 200–220.
[14]Han, H. and Huang, Z., Tailored finite point method for steady-state reaction-diffusion equations, Commun. Math. Sci., 8 (2010), 887–899.
[15]Han, H., Huang, Z., and Kellogg, R. B., The tailored finite point method and a problem of Hemker, P., in Proceedings of International Conference on Boundary and Interior Layers – Computational and Asymptotic Methods, 2008.
[16]Han, H., Huang, Z., and Kellogg, R. B., A tailored finite point method for a singular perturbation problem on an unbounded domain, Journal of Scientific Computing, 36 (2008), 243–261.
[17]Hsieh, P.-W., Shih, Y. and Yang, S.-Y., A tailored finite point method for solving steady MHD duct flow problems with boundary layers, Commun. Comput. Phys., 10 (2011), 161–182.
[18]Huang, Z. and Yang, X., Tailored finite cell method for solving Helmholtz equation in layered heterogeneous medium, J. Comput. Math., 30(4), (2012),381–391.
[19]Larsen, E. W., Morel, J. E., and Miller Jr., W. F., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys., 69 (1987), 283–324.
[20]Larsen, E. W. and Morel, J. E., Asymptotic solutions of numerical transport problems in optically thick,diffusive regimes II, J. Comput. Phys., 83 (1989), 212–236.
[21]Larsen, E. W., The asymptotic diffusion limit of discretized transport problems. Nucl. Sci. Eng., 112 (1992), 336–346, (review).
[22]Larsen, E. W. and Morel, J. E., Advances in Discrete-Ordinates Methodology, in Nuclear Computational Science: A Century in Review, edited by Azmy, Y. Y. and Sartori, E., Springer-Verlag, Berlin. (2010).
[23]Lawrence, R. D., Progress in nodal methods for the solution of the neutron diffusion and transport equations. Prog. Nucl. Energy, 17, (1986), 271 (review).
[24]Lewis, E. E. and Jr, W.F. Miller. Computational Methods of Neutron Transport. John Wiley and Sons, New York, (1984).
[25]Lemou, M. and Mehats, F., Micro-macro schemes for kinetic equations including boundary layers, SIAM J. Sci. Compt. 34(6) (2012), 734–760.
[26]Shih, Y., Kellogg, R. B., and Chang, Y., Characteristic tailored finite point method for convection-dominated convection-diffusion-reaction problems, Journal of Scientific Computing, 47 (2011), 198–215.
[27]Shih, Y., Kellogg, R. B., and Tsai, P., A tailored finite point method for convection-diffusion-reaction problems, Journal of Scientific Computing, 43 (2010), 239–260.
[28]Tang, M., A uniform first order method for the discrete ordinate transport equation with interfaces in X,Y-geometry. Journal of Computational Mathematics, 27 (2009), 764–786.
[29]Warsa, J.S., Wareing, T. A., Morel, J.E., Fully consistent diffusion synthetic acceleration of linear discontinuous SN transport discretizations on unstructured tetrahedral meshes. Nucl. Sci. Eng., 141 (2002), 236–251.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed