{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:36Z","timestamp":1787337216168,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The radiative transfer equation (RTE) occurs in a wide variety of applications. In this paper, we study discrete-ordinate discontinuous Galerkin methods for solving the RTE. The numerical methods are formed in two steps. In the first step, the discrete ordinate technique is applied to discretize the integral operator for the angular variable, resulting in a semidiscrete hyperbolic system. In the second step, the spatial discontinuous Galerkin method is applied to discretize the semidiscrete system. A stability and error analysis is performed on the numerical methods. Some numerical examples are included to demonstrate the convergence behavior of the methods.<\/jats:p>","DOI":"10.1137\/090767340","type":"journal-article","created":{"date-parts":[[2010,2,17]],"date-time":"2010-02-17T18:07:17Z","timestamp":1266430037000},"page":"477-497","source":"Crossref","is-referenced-by-count":50,"title":["Discrete-Ordinate Discontinuous Galerkin Methods for Solving the Radiative Transfer Equation"],"prefix":"10.1137","volume":"32","author":[{"given":"Weimin","family":"Han","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianguo","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joseph A.","family":"Eichholz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,2,17]]},"reference":[{"key":"R1","unstructured":"R. A. Adams,\n                      Sobolev Spaces\n                      , Academic Press, New York, 1975."},{"key":"R2","doi-asserted-by":"crossref","unstructured":"A. Agoshkov,\n                      Boundary Value Problems for Transport Equations\n                      , Birkh\u00e4user, Boston, 1998.","DOI":"10.1007\/978-1-4612-1994-1"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901384162"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2005.11.039"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1017\/S0334270000000278"},{"key":"R6","unstructured":"K. Atkinson and W. Han,\n                      Theoretical Numerical Analysis: A Functional Analysis Framwork\n                      , 3rd ed., Springer, New York, 2009."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/050647177"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , 3rd ed., Springer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.06.015"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202504003866"},{"key":"R11","unstructured":"B. G. Carlson and K. D. Lathrop,\n                      Transport theory\u2014the method of discrete ordinates\n                      , in Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, eds., Gordon and Breach, New York, 1968, pp. 171\u2013266."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/060661363"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.200310088"},{"key":"R15","unstructured":"B. Cockburn,\n                      Discontinuous Galerkin methods for computational fluid dynamics\n                      , in Encyclopedia of Computational Mechanics, Vol. 3, E. Stein, R. de Borst, and T. J. R. Hughes, eds., John Wiley & Sons, New York, 2004, pp. 91\u2013127."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"B. Cockburn, G. E. Karniadakis, and C.W. Shu (eds.),\n                      Discontinuous Galerkin methods. Theory, Computation and Applications\n                      , Lect. Notes Comput. Sci. Eng. 11, Springer, New York, 2000.","DOI":"10.1007\/978-3-642-59721-3"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144503438718"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"W. Freeden, T. Gervens, and M. Schreiner,\n                      Constructive Approximation on the Sphere with Applications to Geomathematics\n                      , Oxford University Press, Oxford, 1998.","DOI":"10.1093\/oso\/9780198536826.001.0001"},{"key":"R19","unstructured":"W. Han, J. A. Eichholz, J. Huang, and J. Lu,\n                      RTE based bioluminescence tomography: A theoretical study\n                      , submitted."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1086\/144246"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2006.01.004"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0815828-4"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500382312"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"P. Lesaint and P. A. Raviart,\n                      On a finite element method for solving the neutron transport equation\n                      , in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. A. de Boor, ed., Academic Press, New York, 1974, pp. 89\u2013123.","DOI":"10.1016\/B978-0-12-208350-1.50008-X"},{"key":"R25","unstructured":"E. E. Lewis and W. F. Miller,\n                      Computational Methods of Neutron Transport\n                      , John Wiley & Sons, New York, 1984."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.08.020"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"M. F. Modest,\n                      Radiative Heat Transfer\n                      , 2nd ed., Academic Press, New York, 2003.","DOI":"10.1016\/B978-012503163-9\/50023-0"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"F. Natterer and F. W\u00fcbbeling,\n                      Mathematical Methods in Image Reconstruction\n                      , SIAM, Philadelphia, 2001.","DOI":"10.1137\/1.9780898718324"},{"key":"R29","unstructured":"W. H. Reed and T. R. Hill,\n                      Triangular Mesh Methods for the Neutron Transport Equation\n                      , Technical report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM, 1973."},{"key":"R30","unstructured":"A. H. Stroud,\n                      Approximate Calculation of Multiple Integrals\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1971."},{"key":"R31","doi-asserted-by":"crossref","unstructured":"G. E. Thomas and K. Stamnes,\n                      Radiative Transfer in the Atmosphere and Ocean\n                      , Cambridge University Press, London, 1999.","DOI":"10.1017\/CBO9780511613470"},{"key":"R32","doi-asserted-by":"crossref","unstructured":"W. Zdunkowski, T. Trautmann, and A. Bott,\n                      Radiation in the Atmosphere: A Course in Theoretical Meteorology\n                      , Cambridge University Press, London, 2007.","DOI":"10.1017\/CBO9780511535796"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090767340","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:49:58Z","timestamp":1787334598000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090767340"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090767340"],"URL":"https:\/\/doi.org\/10.1137\/090767340","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}