{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:15:51Z","timestamp":1776867351465,"version":"3.51.2"},"reference-count":35,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This work deals with the first Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with relaxation. The model problem is written as a <jats:inline-formula id=\"j_cmam-2018-0006_ineq_9999_w2aab3b7e3977b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>P<\/m:mi>\n                              <m:mi>N<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0006_eq_1349.png\"\/>\n                        <jats:tex-math>{P_{N}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> or <jats:inline-formula id=\"j_cmam-2018-0006_ineq_9998_w2aab3b7e3977b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>S<\/m:mi>\n                              <m:mi>N<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0006_eq_1364.png\"\/>\n                        <jats:tex-math>{S_{N}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> model, and we study in more details the <jats:inline-formula id=\"j_cmam-2018-0006_ineq_9997_w2aab3b7e3977b1b6b1aab1c14b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>P<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0006_eq_1348.png\"\/>\n                        <jats:tex-math>{P_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> model in dimension 1 and 2. We show that the TDG method provides natural well-balanced and asymptotic preserving discretization since exact solutions are used locally in the basis functions. High-order convergence with respect to the mesh size in two dimensions is proved together with the asymptotic property for <jats:inline-formula id=\"j_cmam-2018-0006_ineq_9996_w2aab3b7e3977b1b6b1aab1c14b1b7Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>P<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0006_eq_1348.png\"\/>\n                        <jats:tex-math>{P_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> model\nin dimension one. Numerical results in dimensions 1 and 2 illustrate the theoretical properties.<\/jats:p>","DOI":"10.1515\/cmam-2018-0006","type":"journal-article","created":{"date-parts":[[2018,3,22]],"date-time":"2018-03-22T22:15:52Z","timestamp":1521756952000},"page":"521-557","source":"Crossref","is-referenced-by-count":7,"title":["Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the <i>P<\/i>\n                  <sub>1<\/sub> Model"],"prefix":"10.1515","volume":"18","author":[{"given":"Guillaume","family":"Morel","sequence":"first","affiliation":[{"name":"CEA , DAM , DIF , 91297 Arpajon ; and Laboratoire Jacques-Louis Lions, UMR 7598, Universit\u00e9 Pierre et Marie Curie (Paris VI), 75005 Paris , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Buet","sequence":"additional","affiliation":[{"name":"CEA , DAM , DIF , 91297 Arpajon , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruno","family":"Despres","sequence":"additional","affiliation":[{"name":"Laboratoire Jacques-Louis Lions, UMR 7598 , Universit\u00e9 Pierre et Marie Curie (Paris VI) , 75005 Paris ; and Institut Universitaire de France , France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2018,3,22]]},"reference":[{"key":"2023033110284086955_j_cmam-2018-0006_ref_001_w2aab3b7e3977b1b6b1ab2b2b1Aa","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold,\nAn interior penalty finite element method with discontinuous elements,\nSIAM J. Numer. Anal. 19 (1982), 742\u2013760.","DOI":"10.1137\/0719052"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_002_w2aab3b7e3977b1b6b1ab2b2b2Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner,\nPoincar\u00e9\u2013Friedrichs inequalities for piecewise H1H^{1} functions,\nSIAM J. Numer. Anal. 41 (2003), no. 1, 306\u2013324.","DOI":"10.1137\/S0036142902401311"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_003_w2aab3b7e3977b1b6b1ab2b2b3Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_004_w2aab3b7e3977b1b6b1ab2b2b4Aa","doi-asserted-by":"crossref","unstructured":"C.  Buet, B.  Despr\u00e9s and E.  Franck,\nAsymptotic preserving schemes on distorted meshes for Friedrichs systems with stiff relaxation: Application to angular models in linear transport,\nJ. Sci. Comput. 62 (2015), no. 2, 371\u2013398.","DOI":"10.1007\/s10915-014-9859-4"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_005_w2aab3b7e3977b1b6b1ab2b2b5Aa","doi-asserted-by":"crossref","unstructured":"C.  Buet, B.  Despr\u00e9s, E.  Franck and T.  Leroy,\nProof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes,\nMath. Comp. 86 (2017), no. 305, 1147\u20131202.","DOI":"10.1090\/mcom\/3131"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_006_w2aab3b7e3977b1b6b1ab2b2b6Aa","doi-asserted-by":"crossref","unstructured":"A.  Buffa and P.  Monk,\nError estimates for the ultra weak variational formulation of the Helmholtz equation,\nESAIM Math. Model. Numer. Anal. 42 (2008), no. 6, 925\u2013940.","DOI":"10.1051\/m2an:2008033"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_007_w2aab3b7e3977b1b6b1ab2b2b7Aa","doi-asserted-by":"crossref","unstructured":"O.  Cessenat and B.  Despres,\nApplication of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,\nSIAM J. Numer. Anal. 35 (1998), no. 1, 255\u2013299.","DOI":"10.1137\/S0036142995285873"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_008_w2aab3b7e3977b1b6b1ab2b2b8Aa","unstructured":"S.  Chandrasekhar,\nRadiative Transfer,\nInternat. Ser. Monogr. Phys.,\nClarendon Press, Oxford, 1950."},{"key":"2023033110284086955_j_cmam-2018-0006_ref_009_w2aab3b7e3977b1b6b1ab2b2b9Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nDiscontinuous Galerkin methods for Friedrichs\u2019 systems,\nNumerical Mathematics and Advanced Applications (Santiago de Compostela 2005),\nSpringer, Berlin (2006), 79\u201396.","DOI":"10.1007\/978-3-540-34288-5_5"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_010_w2aab3b7e3977b1b6b1ab2b2c10Aa","doi-asserted-by":"crossref","unstructured":"W.  Fleming,\nFunctions of Several Variables, 2nd ed.,\nUndergrad. Texts Math.,\nSpringer, New York, 1977.","DOI":"10.1007\/978-1-4684-9461-7"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_011_w2aab3b7e3977b1b6b1ab2b2c11Aa","doi-asserted-by":"crossref","unstructured":"K. O.  Friedrichs,\nSymmetric positive linear differential equations,\nComm. Pure Appl. Math. 11 (1958), no. 3, 333\u2013418.","DOI":"10.1002\/cpa.3160110306"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_012_w2aab3b7e3977b1b6b1ab2b2c12Aa","doi-asserted-by":"crossref","unstructured":"G.  Gabard,\nDiscontinuous Galerkin methods with plane waves for the displacement-based acoustic equation,\nInternat. J. Numer. Methods Engrg. 66 (2006), 549\u2013569.","DOI":"10.1002\/nme.1571"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_013_w2aab3b7e3977b1b6b1ab2b2c13Aa","doi-asserted-by":"crossref","unstructured":"G.  Gabard,\nDiscontinuous Galerkin methods with plane waves for time-harmonic problems,\nJ. Comput. Phys. 225 (2007), no. 2, 1961\u20131984.","DOI":"10.1016\/j.jcp.2007.02.030"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_014_w2aab3b7e3977b1b6b1ab2b2c14Aa","doi-asserted-by":"crossref","unstructured":"C. J.  Gittelson, R.  Hiptmair and I.  Perugia,\nPlane wave discontinuous Galerkin methods: Analysis of the h-version,\nESAIM Math. Model. Numer. Anal. 43 (2009), no. 2, 297\u2013331.","DOI":"10.1051\/m2an\/2009002"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_015_w2aab3b7e3977b1b6b1ab2b2c15Aa","doi-asserted-by":"crossref","unstructured":"L.  Gosse,\nComputing Qualitatively Correct Approximations of Balance Laws. Exponential-fit, Well-Balanced and Asymptotic-Preserving,\nSpringer, Milano, 2013.","DOI":"10.1007\/978-88-470-2892-0"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_016_w2aab3b7e3977b1b6b1ab2b2c16Aa","doi-asserted-by":"crossref","unstructured":"L.  Gosse and G.  Toscani,\nAn asymptotic-preserving well-balanced scheme for the hyperbolic heat equations,\nC. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 337\u2013342.","DOI":"10.1016\/S1631-073X(02)02257-4"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_017_w2aab3b7e3977b1b6b1ab2b2c17Aa","doi-asserted-by":"crossref","unstructured":"F.  Hermeline,\nA discretization of the multigroup PNP_{N} radiative transfer equation on general meshes,\nJ. Comput. Phys. 313 (2016), 549\u2013582.","DOI":"10.1016\/j.jcp.2016.02.058"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_018_w2aab3b7e3977b1b6b1ab2b2c18Aa","doi-asserted-by":"crossref","unstructured":"R.  Hiptmair, A.  Moiola and I.  Perugia,\nPlane wave discontinuous Galerkin methods for the 2D Helmholtz equation: Analysis of the p-version,\nSIAM J. Numer. Anal. 49 (2011), no. 1, 264\u2013284.","DOI":"10.1137\/090761057"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_019_w2aab3b7e3977b1b6b1ab2b2c19Aa","doi-asserted-by":"crossref","unstructured":"R.  Hiptmair, A.  Moiola and I.  Perugia,\nA survey of Trefftz methods for the Helmholtz equation,\nBuilding Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations,\nSpringer, Cham (2016), 237\u2013278.","DOI":"10.1007\/978-3-319-41640-3_8"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_020_w2aab3b7e3977b1b6b1ab2b2c20Aa","doi-asserted-by":"crossref","unstructured":"R.  Hiptmair, A.  Moiola and I.  Perugia,\nPlane wave discontinuous Galerkin methods: Exponential convergence of the hp-version,\nFound. Comput. Math. 16 (2016), no. 3, 637\u2013675.","DOI":"10.1007\/s10208-015-9260-1"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_021_w2aab3b7e3977b1b6b1ab2b2c21Aa","doi-asserted-by":"crossref","unstructured":"T.  Huttunen, P.  Monk and J. P.  Kaipio,\nComputational aspects of the ultra-weak variational formulation,\nJ. Comput. Phys. 182 (2002), no. 1, 27\u201346.","DOI":"10.1006\/jcph.2002.7148"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_022_w2aab3b7e3977b1b6b1ab2b2c22Aa","doi-asserted-by":"crossref","unstructured":"L.-M.  Imbert-G\u00e9rard,\nInterpolation properties of generalized plane waves,\nNumer. Math. 131 (2015), no. 4, 683\u2013711.","DOI":"10.1007\/s00211-015-0704-y"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_023_w2aab3b7e3977b1b6b1ab2b2c23Aa","doi-asserted-by":"crossref","unstructured":"L.-M.  Imbert-G\u00e9rard,\nWell-posedness and generalized plane waves simulations of a 2D mode conversion model,\nJ. Comput. Phys. 303 (2015), 105\u2013124.","DOI":"10.1016\/j.jcp.2015.09.033"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_024_w2aab3b7e3977b1b6b1ab2b2c24Aa","doi-asserted-by":"crossref","unstructured":"L.-M.  Imbert-G\u00e9rard and B.  Despr\u00e9s,\nA generalized plane-wave numerical method for smooth nonconstant coefficients,\nIMA J. Numer. Anal. 34 (2014), no. 3, 1072\u20131103.","DOI":"10.1093\/imanum\/drt030"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_025_w2aab3b7e3977b1b6b1ab2b2c25Aa","unstructured":"S.  Jin,\nAsymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review,\nRiv. Math. Univ. Parma (N.S.) 3 (2012), no. 2, 177\u2013216."},{"key":"2023033110284086955_j_cmam-2018-0006_ref_026_w2aab3b7e3977b1b6b1ab2b2c26Aa","doi-asserted-by":"crossref","unstructured":"S.  Jin and C.  Levermore,\nNumerical schemes for hyperbolic conservation laws with stiff relaxation terms,\nJ. Comput. Phys. 126 (1996), no. 2, 449\u2013467.","DOI":"10.1006\/jcph.1996.0149"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_027_w2aab3b7e3977b1b6b1ab2b2c27Aa","doi-asserted-by":"crossref","unstructured":"S.  Jin, M.  Tang and H.  Han,\nA uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface,\nNetw. Heterog. Media 4 (2009), no. 1, 35\u201365.","DOI":"10.3934\/nhm.2009.4.35"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_028_w2aab3b7e3977b1b6b1ab2b2c28Aa","doi-asserted-by":"crossref","unstructured":"F.  Kretzschmar, A.  Moiola, I.  Perugia and S. M.  Schnepp,\nA priori error analysis of space\u2013time Trefftz discontinuous Galerkin methods for wave problems,\nIMA J. Numer. Anal. 36 (2016), no. 4, 1599\u20131635.","DOI":"10.1093\/imanum\/drv064"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_029_w2aab3b7e3977b1b6b1ab2b2c29Aa","doi-asserted-by":"crossref","unstructured":"Q.  Li, J.  Lu and W.  Sun,\nDiffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics,\nJ. Comput. Phys. 292 (2015), 141\u2013167.","DOI":"10.1016\/j.jcp.2015.03.014"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_030_w2aab3b7e3977b1b6b1ab2b2c30Aa","unstructured":"E. A.  Maunder,\nTrefftz in translation,\nComput. Assist. Mech. Eng. Sci. 10 (2003), no. 4, 545\u2013563."},{"key":"2023033110284086955_j_cmam-2018-0006_ref_031_w2aab3b7e3977b1b6b1ab2b2c31Aa","doi-asserted-by":"crossref","unstructured":"J.  Melenk and I.  Babu\u0161ka,\nThe partition of unity finite element method: Basic theory and applications,\nComput. Methods Appl. Mech. Engrg. 139 (1996), no. 1\u20134, 289\u2013314.","DOI":"10.1016\/S0045-7825(96)01087-0"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_032_w2aab3b7e3977b1b6b1ab2b2c32Aa","doi-asserted-by":"crossref","unstructured":"P.  Monk and G. R.  Richter,\nA discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media,\nJ. Sci. Comput. 22-23 (2005), 443\u2013477.","DOI":"10.1007\/s10915-004-4132-5"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_033_w2aab3b7e3977b1b6b1ab2b2c33Aa","doi-asserted-by":"crossref","unstructured":"J.  Ragusa, J.-L.  Guermond and G.  Kanschat,\nA robust SNS_{N}-DG-approximation for radiation transport in optically thick and diffusive regimes,\nJ. Comput. Phys. 231 (2012), no. 4, 1947\u20131962.","DOI":"10.1016\/j.jcp.2011.11.017"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_034_w2aab3b7e3977b1b6b1ab2b2c34Aa","doi-asserted-by":"crossref","unstructured":"M.  Tang,\nA uniform first-order method for the discrete ordinate transport equation with interfaces in X,Y-geometry,\nJ. Comput. Math. 27 (2009), no. 6, 764\u2013786.","DOI":"10.4208\/jcm.2009.09-m2894"},{"key":"2023033110284086955_j_cmam-2018-0006_ref_035_w2aab3b7e3977b1b6b1ab2b2c35Aa","doi-asserted-by":"crossref","unstructured":"L.  Wu and Y.  Guo,\nGeometric correction for diffusive expansion of steady neutron transport equation,\nCommun. Math. Phys. 336 (2015), no. 3, 1473\u20131553.","DOI":"10.1007\/s00220-015-2315-y"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/3\/article-p521.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0006\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0006\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:58:09Z","timestamp":1680263889000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0006\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,22]]},"references-count":35,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,5,16]]},"published-print":{"date-parts":[[2018,7,1]]}},"alternative-id":["10.1515\/cmam-2018-0006"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0006","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2018,3,22]]}}}