{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T19:30:24Z","timestamp":1769283024997,"version":"3.49.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T00:00:00Z","timestamp":1621900800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T00:00:00Z","timestamp":1621900800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,7]]},"DOI":"10.1007\/s10915-021-01530-y","type":"journal-article","created":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T12:03:10Z","timestamp":1621944190000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Spherical Harmonic Discontinuous Galerkin Method for Radiative Transfer Equations with Vacuum Boundary Conditions"],"prefix":"10.1007","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4637-8856","authenticated-orcid":false,"given":"Qiwei","family":"Sheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheng","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,5,25]]},"reference":[{"issue":"3","key":"1530_CR1","doi-asserted-by":"publisher","first-page":"298","DOI":"10.13182\/NSE00-41","volume":"137","author":"ML Adams","year":"2001","unstructured":"Adams, M.L.: Discontinuous finite element transport solutions in thick diffusive problems. Nuclear Sci. Eng. 137(3), 298\u2013333 (2001)","journal-title":"Nuclear Sci. Eng."},{"key":"1530_CR2","volume-title":"Sobolev Spaces. Pure and Applied Mathematics","author":"RA Adams","year":"2003","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics. Academic Press, Cambridge (2003)"},{"key":"1530_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1994-1","volume-title":"Boundary Value Problems for Transport Equations Modeling and Simulation in Science, Engineering and Technology","author":"V Agoshkov","year":"1998","unstructured":"Agoshkov, V.: Boundary Value Problems for Transport Equations Modeling and Simulation in Science, Engineering and Technology. Birkh\u00e4user, Boston (1998)"},{"key":"1530_CR4","unstructured":"Atkinson, K., Han, W.: Theoretical numerical analysis: A functional analysis framework. Texts in Applied Mathematics, vol. 39, 3rd edn. Springer, New York (2009)"},{"key":"1530_CR5","doi-asserted-by":"crossref","unstructured":"Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. Lecture Notes in Mathematics, vol. 2044. Springer Science & Business Media, Berlin (2012)","DOI":"10.1007\/978-3-642-25983-8"},{"issue":"2","key":"1530_CR6","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1006\/jcph.1994.1159","volume":"114","author":"JP Berenger","year":"1994","unstructured":"Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185\u2013200 (1994). https:\/\/doi.org\/10.1006\/jcph.1994.1159","journal-title":"J. Comput. Phys."},{"key":"1530_CR7","doi-asserted-by":"crossref","unstructured":"Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol.\u00a015, 3rd edn. Springer, New York (2008)","DOI":"10.1007\/978-0-387-75934-0"},{"issue":"1","key":"1530_CR8","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1016\/j.jcp.2005.04.011","volume":"210","author":"TA Brunner","year":"2005","unstructured":"Brunner, T.A., Holloway, J.P.: Two-dimensional time dependent Riemann solvers for neutron transport. J. Comput. Phys. 210(1), 386\u2013399 (2005). https:\/\/doi.org\/10.1016\/j.jcp.2005.04.011","journal-title":"J. Comput. Phys."},{"key":"1530_CR9","volume-title":"Linear Transport Theory. Addison-Wesley Series in Nuclear Engineering","author":"K Case","year":"1967","unstructured":"Case, K., Zweifel, P.: Linear Transport Theory. Addison-Wesley Series in Nuclear Engineering. Addison-Wesley, Reading, MA (1967)"},{"key":"1530_CR10","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2002). 10.1137\/1.9780898719208","DOI":"10.1137\/1.9780898719208"},{"key":"1530_CR11","volume-title":"Atmospheric Radiation: A Primer with Illustrative Solutions","author":"JA Coakley Jr","year":"2014","unstructured":"Coakley Jr., J.A., Yang, P.: Atmospheric Radiation: A Primer with Illustrative Solutions. Wiley-VCH, Germany (2014)"},{"issue":"11","key":"1530_CR12","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1002\/zamm.200310088","volume":"83","author":"B Cockburn","year":"2003","unstructured":"Cockburn, B.: Discontinuous Galerkin methods. ZAMM - J. Appl. Math. Mech. 83(11), 731\u2013754 (2003). https:\/\/doi.org\/10.1002\/zamm.200310088","journal-title":"ZAMM - J. Appl. Math. Mech."},{"issue":"3","key":"1530_CR13","doi-asserted-by":"publisher","first-page":"1250","DOI":"10.1137\/060677215","volume":"46","author":"B Cockburn","year":"2008","unstructured":"Cockburn, B., Dong, B., Guzm\u00e1n, J.: Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46(3), 1250\u20131265 (2008). https:\/\/doi.org\/10.1137\/060677215","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1530_CR14","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1137\/S0036142900371544","volume":"39","author":"B Cockburn","year":"2001","unstructured":"Cockburn, B., Kanschat, G., Perugia, I., Sch\u00f6tzau, D.: Superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids. SIAM J. Numer. Anal. 39(1), 264\u2013285 (2001). https:\/\/doi.org\/10.1137\/S0036142900371544","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1530_CR15","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/j.atmosres.2004.03.021","volume":"72","author":"P Cook","year":"2004","unstructured":"Cook, P., de Oliveira, C., Haigh, J., Goddard, A.: A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds: Part II. Some applications. Atmosp. Res. 72(1), 223\u2013237 (2004)","journal-title":"Atmosp. Res."},{"key":"1530_CR16","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dry025","author":"K Czuprynski","year":"2018","unstructured":"Czuprynski, K., Eichholz, J., Han, W.: Numerical analysis of the energy-dependent radiative transfer equation. IMA J. Numer. Anal. (2018). https:\/\/doi.org\/10.1093\/imanum\/dry025","journal-title":"IMA J. Numer. Anal."},{"key":"1530_CR17","volume-title":"Mathematical Aspects of Discontinuous Galerkin Methods. Math\u00e9matiques et Applications","author":"D Di Pietro","year":"2011","unstructured":"Di Pietro, D., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Math\u00e9matiques et Applications. Springer, Berlin (2011)"},{"key":"1530_CR18","volume-title":"Transport Theory","author":"JJ Duderstadt","year":"1978","unstructured":"Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1978)"},{"issue":"03","key":"1530_CR19","doi-asserted-by":"publisher","first-page":"1150014","DOI":"10.1142\/S021820251150014X","volume":"22","author":"H Egger","year":"2012","unstructured":"Egger, H., Schlottbom, M.: A mixed variational framework for the radiative transfer equation. Math. Models Methods Appl. Sci. 22(03), 1150014 (2012). https:\/\/doi.org\/10.1142\/S021820251150014X","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3\u20134","key":"1530_CR20","doi-asserted-by":"publisher","first-page":"365","DOI":"10.3233\/ASY-141235","volume":"89","author":"H Egger","year":"2014","unstructured":"Egger, H., Schlottbom, M.: Diffusion asymptotics for linear transport with low regularity. Asymptot. Anal. 89(3\u20134), 365\u2013377 (2014). https:\/\/doi.org\/10.3233\/ASY-141235","journal-title":"Asymptot. Anal."},{"issue":"6","key":"1530_CR21","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1080\/00036811.2013.826798","volume":"93","author":"H Egger","year":"2014","unstructured":"Egger, H., Schlottbom, M.: An $${L}^p$$ theory for stationary radiative transfer. Appl. Anal. 93(6), 1283\u20131296 (2014). https:\/\/doi.org\/10.1080\/00036811.2013.826798","journal-title":"Appl. Anal."},{"issue":"5","key":"1530_CR22","doi-asserted-by":"publisher","first-page":"2166","DOI":"10.1137\/18M1172521","volume":"57","author":"H Egger","year":"2019","unstructured":"Egger, H., Schlottbom, M.: A perfectly matched layer approach for $${P}_{N}$$-approximations in radiative transfer. SIAM J. Numer. Anal. 57(5), 2166\u20132188 (2019). https:\/\/doi.org\/10.1137\/18M1172521","journal-title":"SIAM J. Numer. Anal."},{"key":"1530_CR23","doi-asserted-by":"crossref","unstructured":"Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer-Verlag, New York (2004)","DOI":"10.1007\/978-1-4757-4355-5"},{"issue":"5","key":"1530_CR24","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.4310\/CMS.2016.v14.n5.a10","volume":"14","author":"M Frank","year":"2016","unstructured":"Frank, M., Hauck, C., K\u00fcpper, K.: Convergence of filtered spherical harmonic equations for radiation transport. Commun. Math. Sci. 14(5), 1443\u20131465 (2016). https:\/\/doi.org\/10.4310\/CMS.2016.v14.n5.a10","journal-title":"Commun. Math. Sci."},{"issue":"2","key":"1530_CR25","doi-asserted-by":"publisher","first-page":"2289","DOI":"10.1016\/j.jcp.2007.07.009","volume":"226","author":"M Frank","year":"2007","unstructured":"Frank, M., Klar, A., Larsen, E.W., Yasuda, S.: Time-dependent simplified $${P}_{N}$$ approximation to the equations of radiative transfer. J. Comput. Phys. 226(2), 2289\u20132305 (2007). https:\/\/doi.org\/10.1016\/j.jcp.2007.07.009","journal-title":"J. Comput. Phys."},{"issue":"3","key":"1530_CR26","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1080\/00411450903187722","volume":"38","author":"H Gao","year":"2009","unstructured":"Gao, H., Zhao, H.: A fast-forward solver of radiative transfer equation. Transp. Theory Stat. Phys. 38(3), 149\u2013192 (2009). https:\/\/doi.org\/10.1080\/00411450903187722","journal-title":"Transp. Theory Stat. Phys."},{"issue":"281","key":"1530_CR27","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1090\/S0025-5718-2012-02605-6","volume":"82","author":"H Gao","year":"2013","unstructured":"Gao, H., Zhao, H.: Analysis of a numerical solver for radiative transport equation. Math. Comput. 82(281), 153\u2013172 (2013). https:\/\/doi.org\/10.1090\/S0025-5718-2012-02605-6","journal-title":"Math. Comput."},{"issue":"2","key":"1530_CR28","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.camwa.2015.05.030","volume":"72","author":"CK Garrett","year":"2016","unstructured":"Garrett, C.K., Hauck, C.D.: On the eigenstructure of spherical harmonic equations for radiative transport. Comput. Math. Appl. 72(2), 264\u2013270 (2016). https:\/\/doi.org\/10.1016\/j.camwa.2015.05.030","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"1530_CR29","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1137\/090767340","volume":"32","author":"W Han","year":"2010","unstructured":"Han, W., Huang, J., Eichholz, J.: Discrete-ordinate discontinuous Galerkin methods for solving the radiative transfer equation. SIAM J. Sci. Comput. 32(2), 477\u2013497 (2010). https:\/\/doi.org\/10.1137\/090767340","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1530_CR30","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1137\/1025002","volume":"25","author":"A Harten","year":"1983","unstructured":"Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35\u201361 (1983). https:\/\/doi.org\/10.1137\/1025002","journal-title":"SIAM Rev."},{"key":"1530_CR31","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1086\/144246","volume":"93","author":"L Henyey","year":"1941","unstructured":"Henyey, L., Greenstein, J.: Diffuse radiation in the galaxy. Astrophys. J. 93, 70\u201383 (1941)","journal-title":"Astrophys. J."},{"issue":"2","key":"1530_CR32","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/j.cma.2004.04.009","volume":"194","author":"P Houston","year":"2005","unstructured":"Houston, P., S\u00fcli, E.: A note on the design of $$hp$$-adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(2), 229\u2013243 (2005). https:\/\/doi.org\/10.1016\/j.cma.2004.04.009","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"173","key":"1530_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008211","volume":"46","author":"C Johnson","year":"1986","unstructured":"Johnson, C., Pitk\u00e4ranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46(173), 1\u201326 (1986). https:\/\/doi.org\/10.2307\/2008211","journal-title":"Math. Comput."},{"issue":"4","key":"1530_CR34","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1515\/cmam-2016-0023","volume":"16","author":"G Kanschat","year":"2016","unstructured":"Kanschat, G., Lorca, J.P.L.: A weakly penalized discontinuous Galerkin method for radiation in dense, scattering media. Computat. Methods Appl. Math. 16(4), 563\u2013577 (2016). https:\/\/doi.org\/10.1515\/cmam-2016-0023","journal-title":"Computat. Methods Appl. Math."},{"key":"1530_CR35","doi-asserted-by":"publisher","unstructured":"Laiu, M., Hauck, C., McClarren, R., O\u2019Leary, D., Tits, A.: Positive filtered $${P}_{N}$$ moment closures for linear kinetic equations. SIAM J. Numer. Anal. 54(6), 3214\u20133238 (2016). https:\/\/doi.org\/10.1137\/15M1052871","DOI":"10.1137\/15M1052871"},{"key":"1530_CR36","doi-asserted-by":"crossref","unstructured":"Lasaint, P., Raviart, P.: On a finite element method for solving the neutron transport equation. In: C.\u00a0de Boor (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89 \u2013 123. Academic Press (1974)","DOI":"10.1016\/B978-0-12-208350-1.50008-X"},{"key":"1530_CR37","volume-title":"Computational Methods of Neutron Transport","author":"EE Lewis","year":"1984","unstructured":"Lewis, E.E., Miller, W.F.: Computational Methods of Neutron Transport. Wiley, New York (1984)"},{"key":"1530_CR38","doi-asserted-by":"crossref","unstructured":"Li, Q., Lu, J., Sun, W.: Diffusion approximations and domain decomposition method of linear transport equations: asymptotics and numerics. J. Comput. Phys. 292(Supplement C), 141\u2013167 (2015)","DOI":"10.1016\/j.jcp.2015.03.014"},{"issue":"2","key":"1530_CR39","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1137\/S0036142998344706","volume":"37","author":"T Manteuffel","year":"1999","unstructured":"Manteuffel, T., Ressel, K., Starke, G.: A boundary functional for the least-squares finite-element solution of neutron transport problems. SIAM J. Numer. Anal. 37(2), 556\u2013586 (1999). https:\/\/doi.org\/10.1137\/S0036142998344706","journal-title":"SIAM J. Numer. Anal."},{"key":"1530_CR40","volume-title":"Radiative Heat Transfer","author":"MF Modest","year":"2013","unstructured":"Modest, M.F.: Radiative Heat Transfer, 3rd edn. Academic Press, Oxford (2013)","edition":"3"},{"key":"1530_CR41","doi-asserted-by":"crossref","unstructured":"M\u00fcller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)","DOI":"10.1007\/BFb0094775"},{"key":"1530_CR42","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139164474","volume-title":"An Introduction to Radiative Transfer: Methods and Applications in Astrophysics","author":"A Peraiah","year":"2001","unstructured":"Peraiah, A.: An Introduction to Radiative Transfer: Methods and Applications in Astrophysics. Cambridge University Press, Cambridge (2001)"},{"key":"1530_CR43","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.jcp.2019.01.024","volume":"384","author":"S Powell","year":"2019","unstructured":"Powell, S., Cox, B.T., Arridge, S.R.: A pseudospectral method for solution of the radiative transport equation. J. Comput. Phys. 384, 376\u2013382 (2019). https:\/\/doi.org\/10.1016\/j.jcp.2019.01.024","journal-title":"J. Comput. Phys."},{"key":"1530_CR44","unstructured":"Reed, W., Hill, T.: Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Lab., N.Mex. (USA) (1973)"},{"key":"1530_CR45","doi-asserted-by":"publisher","unstructured":"Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, Berlin (2011). https:\/\/doi.org\/10.1007\/978-3-540-71041-7","DOI":"10.1007\/978-3-540-71041-7"},{"key":"1530_CR46","doi-asserted-by":"crossref","unstructured":"Thomas, G.E., Stamnes, K.: Radiative Transfer in the Atmosphere and Ocean. Cambridge University Press, Cambridge (1999)","DOI":"10.1017\/CBO9780511613470"},{"issue":"5","key":"1530_CR47","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.4208\/cicp.310715.290316a","volume":"20","author":"C Wang","year":"2016","unstructured":"Wang, C., Sheng, Q., Han, W.: A discrete-ordinate discontinuous-streamline diffusion method for the radiative transfer equation. Commun. Comput. Phys. 20(5), 1443\u20131465 (2016). https:\/\/doi.org\/10.4208\/cicp.310715.290316a","journal-title":"Commun. Comput. Phys."},{"key":"1530_CR48","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511535796","volume-title":"Radiation in the Atmosphere: A Course in Theoretical Meteorology","author":"W Zdunkowski","year":"2007","unstructured":"Zdunkowski, W., Trautmann, T., Bott, A.: Radiation in the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, Cambridge (2007)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01530-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01530-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01530-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,23]],"date-time":"2021-06-23T07:23:51Z","timestamp":1624433031000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01530-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,25]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["1530"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01530-y","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,25]]},"assertion":[{"value":"5 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"9"}}