Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T02:10:20.982Z Has data issue: false hasContentIssue false
  • English
  • Français

A Discrete-Ordinate Discontinuous-Streamline Diffusion Method for the Radiative Transfer Equation

Part of: Partial differential equations, boundary value problems Integral equations, integral transforms

Published online by Cambridge University Press:  02 November 2016

Cheng Wang ,
Qiwei Sheng  and
Weimin Han
Cheng Wang*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
Qiwei Sheng*
Affiliation:
Department of Mathematics, California State University, Bakersfield, CA 93311, USA
Weimin Han*
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
*
*Corresponding author. Email addresses:[email protected] (Q. Sheng), [email protected] (C. Wang), [email protected] (W. Han)
*Corresponding author. Email addresses:[email protected] (Q. Sheng), [email protected] (C. Wang), [email protected] (W. Han)
*Corresponding author. Email addresses:[email protected] (Q. Sheng), [email protected] (C. Wang), [email protected] (W. Han)
Get access

Abstract

The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the spatial discretization. Stability and error estimates in certain norms are proved. Numerical results show that the proposed method can lead to a more accurate approximation in comparison with the DODG method.

Keywords

Radiative transfer equationdiscrete-ordinate methoddiscontinuous-streamline diffusion methodstabilityerror estimation

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65R20: Integral equations
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1443 - 1465
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Agoshkov, V. Boundary Value Problems for Transport Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, 1998.Google Scholar
[2] Arridge, S. R. and Schotland, J. C. Optical tomography: forward and inverse problems. Inverse Problems 25, 12 (2009), 123010.Google Scholar
[3] Asadzadeh, M. and Kazemi, E. On convergence of the streamline diffusion and discontinuous galerkin methods for the multi-dimensional fermi pencil beam equation. Int. J. Numer. Anal. Mod. 10, 4 (2013), 860–875.Google Scholar
[4] Asadzadeh, M. and Kowalczyk, P. Convergence analysis of the streamline diffusion and discontinuous galerkin methods for the vlasov-fokker-planck system. Numerical Methods for Partial Differential Equations 21, 3 (2005), 472–495.Google Scholar
[5] Atkinson, K. An Introduction to Numerical Analysis, 2nd ed. John Wiley & Sons, 1989.Google Scholar
[6] Bal, G. Inverse transport theory and applications. Inverse Problems 25, 5 (2009), 053001.Google Scholar
[7] Brenner, S. and Scott, R. The Mathematical Theory of Finite Element Methods, 3rd ed., vol. 15 of Texts in Applied Mathematics. Springer, 2008.Google Scholar
[8] Carlson, B. G. Solution of the transport equation by the Sn method. Los Alamos National Laboratory (1955).Google Scholar
[9] Sun, C., Tan, H.M., and Wu, K.J. The discontinuous-streamline diffusion method for first-order hyperbolic equation. Mathematica Numerica Sinica 20, 1 (Feb. 1998), 35–44.Google Scholar
[10] Ciarlet, P. The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, 2002.Google Scholar
[11] Egger, H. and Schlottbom, M. Amixed variational framework for the radiative transfer equation. Mathematical Models and Methods in Applied Sciences 22, 03 (2012), 1150014.Google Scholar
[12] Fleck, J. A. Jr. and Cummings, J. D. Jr. An implicit Monte Carlo scheme for calculating time and frequency dependent nonlinear radiation transport. Journal of Computational Physics 8, 3 (1971), 313–342.Google Scholar
[13] Gao, H. and Zhao, H. Analysis of a numerical solver for radiative transport equation. Mathematics of Computation 82, 281 (2013), 153–172.Google Scholar
[14] Han, W., Eichholz, J., Cheng, X., and Wang, G. A theoretical framework of X-ray dark-field tomography. SIAM Journal on Applied Mathematics 71, 5 (2011), 1557–1577.Google Scholar
[15] Han, W., Eichholz, J., and Sheng, Q. Theory of differential approximations of radiative transfer equation. In Advances in Applied Mathematics and Approximation Theory. Springer, 2013, 121–148.Google Scholar
[16] Han, W., Huang, J., and Eichholz, J. Discrete-ordinate discontinuous Galerkin methods for solving the radiative transfer equation. SIAM Journal on Scientific Computing 32, 2 (2010), 477–497.Google Scholar
[17] Han, W., Li, Y., Sheng, Q., and Tang, J. A numerical method for generalized Fokker-Planck equations. Recent Advances in Scientific Computing and Applications 586 (2013), 171–179.Google Scholar
[18] Hesse, K. and Sloan, I. H. Cubature over the sphere S2 in Sobolev spaces of arbitrary order. J. Approx. Theory 141 (2006), 118–133.CrossRefGoogle Scholar
[19] Howell, J. R. The Monte Carlo method in radiative heat transfer. Journal of Heat Transfer 120, 3 (1998), 547–560.Google Scholar
[20] Hughes, T. J., Mallet, M., and Akira, M. A new finite element formulation for computational fluid dynamics: Ii. beyond SUPG. Computer Methods in Applied Mechanics and Engineering 54, 3 (1986), 341–355.Google Scholar
[21] Johnson, C. and Nävert, U. An analysis of some finite element methods for advection-diffusion problems. In Analytical and Numerical Approaches to Asymptotic Problems in Analysis Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems, Axelsson, O., Frank, L., and Sluis, A. V. D., Eds., vol. 47 of North-Holland Mathematics Studies. North-Holland, 1981, 99–116.Google Scholar
[22] Koch, R., Krebs, W., Wittig, S., and Viskanta, R. The discrete ordinate quadrature schemes for multidimensional radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer 53 (1995), 353–372.Google Scholar
[23] Larsen, E. W. and Morel, J. E. Advances in discrete-ordinates methodology. In Nuclear Computational Science. Springer Netherlands, 2010, 1–84.Google Scholar
[24] Ivan, L. and Koblinger, L. Monte Carlo Particle Transport Methods: Neutron and Photon Calculations. CRC Press, 1991.Google Scholar
[25] McClarren, R. G. and Hauck, C. D. Simulating radiative transferwith filtered spherical harmonics. Physics Letters A 374, 22 (2010), 2290–2296.Google Scholar
[26] Ren, K. Recent developments in numerical techniques for transport-based medical imaging methods. Communications in Computational Physics 8 (2010), 1–50.Google Scholar
[27] Spanier, J. and Gelbard, E. Monte Carlo Principles and Neutron Transport Problems. Dover, New York, 2008.Google Scholar
[28] Tang, J., Han, W., and Han, B. A theoretical study for RTE-based parameter identification problems. Inverse Problems 29, 9 (2013), 095002.Google Scholar