{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T19:44:43Z","timestamp":1768074283339,"version":"3.49.0"},"reference-count":53,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"Abstract<\/jats:title>\n In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed.\nOne of its features is that a streamline weight is added to the test function to obtain the variational formulation.\nIn the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space.\nThe use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a C\u00e9a\u2019s type lemma and standard interpolation estimates are sufficient for our analysis.\nFor our numerical scheme, based on \n \n \n \n \u2119<\/m:mi>\n k<\/m:mi>\n <\/m:msup>\n <\/m:math>\n \n {\\mathbb{P}^{k}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> discontinuous Galerkin finite elements (in space) on a mesh of size h<\/jats:italic> and a spherical harmonics approximation of order N<\/jats:italic> (in the angular variable), the convergence rate is of order \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n N<\/m:mi>\n \n -<\/m:mo>\n t<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n \n h<\/m:mi>\n k<\/m:mi>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(N^{-t}+h^{k})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for a smooth solution which admits partial derivatives of order \n \n \n \n k<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {k+1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and t<\/jats:italic> with respect to the spatial and angular variables, respectively. For \n \n \n \n k<\/m:mi>\n =<\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {k=0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (piecewise constant finite elements) we also obtain a convergence result of order \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n N<\/m:mi>\n \n -<\/m:mo>\n t<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n \n h<\/m:mi>\n \n 1<\/m:mn>\n 2<\/m:mn>\n <\/m:mfrac>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(N^{-t}+h^{\\frac{1}{2}})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nNumerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-norm, typically of one more order, \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n N<\/m:mi>\n \n -<\/m:mo>\n t<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n \n h<\/m:mi>\n \n k<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(N^{-t}+h^{k+1})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/cmam-2024-0021","type":"journal-article","created":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T17:03:21Z","timestamp":1727802201000},"page":"287-311","source":"Crossref","is-referenced-by-count":2,"title":["Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0635-7508","authenticated-orcid":false,"given":"Kenneth","family":"Assogba","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Saclay , CEA, Service d\u2019\u00c9tudes des R\u00e9acteurs et de Math\u00e9matiques Appliqu\u00e9es 91191 Gif-sur-Yvette , France"}]},{"given":"Gr\u00e9goire","family":"Allaire","sequence":"additional","affiliation":[{"name":"CMAP , \u00c9cole polytechnique , 52830 Institut Polytechnique de Paris , 91120 Palaiseau , France"}]},{"given":"Lahbib","family":"Bourhrara","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Saclay , CEA, Service d\u2019\u00c9tudes des R\u00e9acteurs et de Math\u00e9matiques Appliqu\u00e9es 91191 Gif-sur-Yvette , France"}]}],"member":"374","published-online":{"date-parts":[[2024,10,2]]},"reference":[{"key":"2025032910202788634_j_cmam-2024-0021_ref_001","unstructured":"R. T. Ackroyd,\nFinite Element Methods for Particle Transport: Applications to Reactor and Radiation Physics,\nRes. Stud. Particle Nuclear Technol. 6,\nResearch Studies Press, Taunton, 1997."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_002","doi-asserted-by":"crossref","unstructured":"G. Allaire,\nNumerical Analysis and Optimization,\nNumer. Math. Sci. Comput.,\nOxford University, Oxford, 2007.","DOI":"10.1093\/oso\/9780199205219.001.0001"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_003","unstructured":"G. Allaire, X. Blanc, B. Despr\u00e9s and F. Golse,\nTransport et diffusion,\nEditions de l\u2019\u00c9cole polytechnique, Palaiseau, 2018."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_004","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,\nUnified analysis of discontinuous Galerkin methods for elliptic problems,\nSIAM J. Numer. Anal. 39 (2001\/02), no. 5, 1749\u20131779.","DOI":"10.1137\/S0036142901384162"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_005","doi-asserted-by":"crossref","unstructured":"M. Asadzadeh,\nAnalysis of a fully discrete scheme for neutron transport in two-dimensional geometry,\nSIAM J. Numer. Anal. 23 (1986), no. 3, 543\u2013561.","DOI":"10.1137\/0723035"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_006","unstructured":"K. Assogba and L. Bourhrara,\nThe PN form of the neutron transport problem achieves linear scalability through domain decomposition,\nProceedings of International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering,\nAmerican Nuclear Society, Westmont (2023)."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_007","doi-asserted-by":"crossref","unstructured":"K. Assogba, L. Bourhrara, I. Zmijarevic and G. Allaire,\nPrecise 3D reactor core calculation using spherical harmonics and discontinuous Galerkin finite element methods,\nProceedings of International Conference on Physics of Reactors 2022,\nAmerican Nuclear Society, Westmont (2022), 1224\u20131233.","DOI":"10.13182\/PHYSOR22-37354"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_008","doi-asserted-by":"crossref","unstructured":"K. Assogba, L. Bourhrara, I. Zmijarevic, G. Allaire and A. Galia,\nSpherical harmonics and discontinuous Galerkin finite element methods for the three-dimensional neutron transport equation: Application to core and lattice calculation,\nNuclear Sci. Eng. 197 (2023), no. 8, 1584\u20131599.","DOI":"10.1080\/00295639.2022.2154546"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_009","doi-asserted-by":"crossref","unstructured":"K. Atkinson and W. Han,\nTheoretical Numerical Analysis, 2nd ed.,\nTexts Appl. 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I Math. 300 (1985), no. 3, 89\u201392."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_016","doi-asserted-by":"crossref","unstructured":"S. Chandrasekhar,\nOn the radiative equilibrium of a stellar atmosphere. II,\nAstrophys. J. 100 (1944), 76\u201386.","DOI":"10.1086\/144639"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_017","doi-asserted-by":"crossref","unstructured":"P. Ciarlet,\nT-coercivity: Application to the discretization of Helmholtz-like problems,\nComput. Math. Appl. 64 (2012), no. 1, 22\u201334.","DOI":"10.1016\/j.camwa.2012.02.034"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_018","doi-asserted-by":"crossref","unstructured":"P. Ciarlet, M. H. Do and F. Madiot,\nA posteriori error estimates for mixed finite element discretizations of the neutron diffusion equations,\nESAIM Math. Model. Numer. Anal. 57 (2023), no. 1, 1\u201327.","DOI":"10.1051\/m2an\/2022078"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_019","doi-asserted-by":"crossref","unstructured":"W. Dahmen, F. Gruber and O. Mula,\nAn adaptive nested source term iteration for radiative transfer equations,\nMath. Comp. 89 (2020), no. 324, 1605\u20131646.","DOI":"10.1090\/mcom\/3505"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_020","unstructured":"W. Dahmen and O. Mula,\nAccuracy controlled schemes for the eigenvalue problem of the radiative transfer equation,\npreprint (2023), https:\/\/arxiv.org\/abs\/2307.07780."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_021","unstructured":"R. Dautray and J.-L. Lions,\nMathematical Analysis and Numerical Methods for Science and Technology. Vol. 6,\nSpringer, Berlin, 1988."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_022","doi-asserted-by":"crossref","unstructured":"B. Davison,\nSpherical-harmonics method for neutron-transport problems in cylindrical geometry,\nCanad. J. 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Anal. 44 (2006), no. 2, 753\u2013778.","DOI":"10.1137\/050624133"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_026","doi-asserted-by":"crossref","unstructured":"A. Ern, M. Vohral\u00edk and M. Zakerzadeh,\nGuaranteed and robust \n \n \n \n L\n 2\n \n \n \n L^{2}\n \n -norm a posteriori error estimates for 1D linear advection problems,\nESAIM Math. Model. Numer. Anal. 55 (2021), S447\u2013S474.","DOI":"10.1051\/m2an\/2020041"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_027","doi-asserted-by":"crossref","unstructured":"A. Gammicchia, S. Santandrea and S. Dulla,\nCross sections polynomial axial expansion within the APOLLO3\u00ae 3D characteristics method,\nAnn. Nuclear Energy 165 (2022), Article ID 108673.","DOI":"10.1016\/j.anucene.2021.108673"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_028","doi-asserted-by":"crossref","unstructured":"K. Grella,\nSparse tensor phase space Galerkin approximation for radiative transport,\nSpringerPlus 3 (2014), Paper No. 230.","DOI":"10.1186\/2193-1801-3-230"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_029","doi-asserted-by":"crossref","unstructured":"K. Grella and C. Schwab,\nSparse tensor spherical harmonics approximation in radiative transfer,\nJ. Comput. Phys. 230 (2011), no. 23, 8452\u20138473.","DOI":"10.1016\/j.jcp.2011.07.028"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_030","doi-asserted-by":"crossref","unstructured":"T. H. Gronwall,\nOn the degree of convergence of Laplace\u2019s series,\nTrans. Amer. Math. Soc. 15 (1914), no. 1, 1\u201330.","DOI":"10.1090\/S0002-9947-1914-1500962-6"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_031","unstructured":"A. H\u00e9bert,\nApplied Reactor Physics,\nPresses internationales Polytechnique, Montreal, 2016."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_032","doi-asserted-by":"crossref","unstructured":"P. Houston, M. E. Hubbard, T. J. Radley, O. J. Sutton and R. S. J. Widdowson,\nEfficient high-order space-angle-energy polytopic discontinuous Galerkin finite element methods for linear Boltzmann transport,\nJ. Sci. Comput. 100 (2024), no. 2, Paper No. 52.","DOI":"10.1007\/s10915-024-02569-3"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_033","unstructured":"C. Johnson,\nNumerical Solution of Partial Differential Equations by the Finite Element Method,\nCambridge University, Cambridge, 1987."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_034","doi-asserted-by":"crossref","unstructured":"C. Johnson, U. N\u00e4vert and J. Pitk\u00e4ranta,\nFinite element methods for linear hyperbolic problems,\nComput. Methods Appl. Mech. Engrg. 45 (1984), no. 1\u20133, 285\u2013312.","DOI":"10.1016\/0045-7825(84)90158-0"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_035","doi-asserted-by":"crossref","unstructured":"C. Johnson and J. Pitk\u00e4ranta,\nConvergence of a fully discrete scheme for two-dimensional neutron transport,\nSIAM J. Numer. Anal. 20 (1983), no. 5, 951\u2013966.","DOI":"10.1137\/0720065"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_036","doi-asserted-by":"crossref","unstructured":"C. Johnson and J. Pitk\u00e4ranta,\nAn analysis of the discontinuous Galerkin method for a scalar hyperbolic equation,\nMath. Comp. 46 (1986), no. 173, 1\u201326.","DOI":"10.1090\/S0025-5718-1986-0815828-4"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_037","unstructured":"P. Lesaint,\nSur la r\u00e9solution des syst\u00e8mes hyperboliques du premier ordre par des m\u00e9thodes d\u2019\u00e9l\u00e9ments finis,\nPhD thesis, Universit\u00e9 Paris VI, 1975."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_038","doi-asserted-by":"crossref","unstructured":"P. Lesaint and P.-A. Raviart,\nOn a finite element method for solving the neutron transport equation,\nMathematical Aspects of Finite Elements in Partial Differential Equations,\nAcademic Press, New York (1974), 89\u2013123.","DOI":"10.1016\/B978-0-12-208350-1.50008-X"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_039","unstructured":"E. E. Lewis and W. F. Miller,\nComputational Methods of Neutron Transport,\nWiley, New York, 1984."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_040","doi-asserted-by":"crossref","unstructured":"T. A. Manteuffel and K. J. Ressel,\nLeast-squares finite-element solution of the neutron transport equation in diffusive regimes,\nSIAM J. Numer. Anal. 35 (1998), no. 2, 806\u2013835.","DOI":"10.1137\/S0036142996299708"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_041","doi-asserted-by":"crossref","unstructured":"E. Masiello, R. Sanchez and I. 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Ressel,\nLeast-squares finite element solution of the neutron transport equation in diffusive regimes,\nPh.D. Thesis, University of Colorado at Denver, 1994."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_048","unstructured":"P. Reuss and J. Bussac,\nTrait\u00e9 de neutronique,\nHermann, Paris, 1978."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_049","doi-asserted-by":"crossref","unstructured":"G. R. Richter,\nAn optimal-order error estimate for the discontinuous Galerkin method,\nMath. Comp. 50 (1988), no. 181, 75\u201388.","DOI":"10.1090\/S0025-5718-1988-0917819-3"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_050","doi-asserted-by":"crossref","unstructured":"R. Sanchez,\nOn PN interface and boundary conditions,\nNuclear Sci. Eng. 177 (2014), no. 1, 19\u201334.","DOI":"10.13182\/NSE12-95"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_051","unstructured":"D. Schneider, F. Dolci, F. Gabriel, J.-M. Palau, M. Guillo and B. Pothet,\nAPOLLO3\u00ae CEA\/DEN deterministic multi-purpose code for reactor physics analysis,\nPHYSOR 2016 \u2013 Unifying Theory and Experiments in the 21st Century,\nCurran Associates, Red Hook (2016), 2274\u20132285."},{"key":"2025032910202788634_j_cmam-2024-0021_ref_052","doi-asserted-by":"crossref","unstructured":"Y. Wang and J. C. Ragusa,\nStandard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes,\nJ. Comput. Phys. 230 (2011), no. 3, 763\u2013788.","DOI":"10.1016\/j.jcp.2010.10.018"},{"key":"2025032910202788634_j_cmam-2024-0021_ref_053","doi-asserted-by":"crossref","unstructured":"G. Widmer, R. Hiptmair and C. Schwab,\nSparse adaptive finite elements for radiative transfer,\nJ. Comput. 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