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

Published online by Cambridge University Press:  03 June 2015

Houde Han ,
Min Tang  and
Wenjun Ying
Houde Han*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Haidian, Beijing 100084, P.R. China
Min Tang*
Affiliation:
Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P.R. China
Wenjun Ying*
Affiliation:
Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P.R. China
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
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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.

Keywords

65L1276N2035Q70Neutron transport equationdiscrete ordinates methodtailored finite point methodboundary layersinterface layers
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014 , pp. 797 - 826
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Adams, M. L., Discontinuous finite element transport solutions in thick diffusive problems, Nucl. Sci. Eng., 137 (2001), 298–333.Google Scholar
[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), 669–680.Google Scholar
[3]Azmy, Y. Y., Arbitrarily high order characteristic methods for solving the neutron transport equation, Ann. Nucl. Energy. 19 (1992), 593–606.CrossRefGoogle Scholar
[4]Bal, G. and Ryzhik, L., Diffusion Approximation of Radiative Transfer Problems with Interfaces, SIAM, J. Appl. Math., 60(6) (2000),1887–1912.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.CrossRefGoogle Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[14]Han, H. and Huang, Z., Tailored finite point method for steady-state reaction-diffusion equations, Commun. Math. Sci., 8 (2010), 887–899.Google Scholar
[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.Google Scholar
[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.CrossRefGoogle Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[21]Larsen, E. W., The asymptotic diffusion limit of discretized transport problems. Nucl. Sci. Eng., 112 (1992), 336–346, (review).Google Scholar
[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).Google Scholar
[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).Google Scholar
[24]Lewis, E. E. and Jr, W.F. Miller. Computational Methods of Neutron Transport. John Wiley and Sons, New York, (1984).Google Scholar
[25]Lemou, M. and Mehats, F., Micro-macro schemes for kinetic equations including boundary layers, SIAM J. Sci. Compt. 34(6) (2012), 734–760.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar