{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:11:35Z","timestamp":1776863495499,"version":"3.51.2"},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["RTG 1294"],"award-info":[{"award-number":["RTG 1294"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CRC 1173"],"award-info":[{"award-number":["CRC 1173"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We introduce a space-time discretization for linear first-order hyperbolic\nevolution systems using a discontinuous Galerkin approximation in space and a\nPetrov\u2013Galerkin scheme in time. We show well-posedness and convergence of the\ndiscrete system. Then we introduce an adaptive strategy based on goal-oriented\ndual-weighted error estimation. The full space-time linear system is solved\nwith a parallel multilevel preconditioner. Numerical experiments for the linear\ntransport equation and the Maxwell equation in 2D underline the efficiency of\nthe overall adaptive solution process.<\/jats:p>","DOI":"10.1515\/cmam-2016-0015","type":"journal-article","created":{"date-parts":[[2016,4,7]],"date-time":"2016-04-07T18:39:22Z","timestamp":1460054362000},"page":"409-428","source":"Crossref","is-referenced-by-count":44,"title":["Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems"],"prefix":"10.1515","volume":"16","author":[{"given":"Willy","family":"D\u00f6rfler","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Angewandte und Numerische Mathematik, KIT, 76049 Karlsruhe, Germany"}]},{"given":"Stefan","family":"Findeisen","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte und Numerische Mathematik, KIT, 76049 Karlsruhe, Germany"}]},{"given":"Christian","family":"Wieners","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte und Numerische Mathematik, KIT, 76049 Karlsruhe, Germany"}]}],"member":"374","published-online":{"date-parts":[[2016,4,7]]},"reference":[{"key":"2023033112444849263_j_cmam-2016-0015_ref_001_w2aab3b7e2264b1b6b1ab2b1b1Aa","unstructured":"Abramowitz M. and Stegun I. A.,\nHandbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables,\nAppl. Math. Ser.,\nDover Publications, New York, 1964."},{"key":"2023033112444849263_j_cmam-2016-0015_ref_002_w2aab3b7e2264b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"Ascher U. M., Ruuth S. J. and Wetton B. T. R.,\nImplicit-explicit methods for time-dependent partial differential equations,\nSIAM J. Numer. Anal. 32 (1995), no. 3, 797\u2013823.","DOI":"10.1137\/0732037"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_003_w2aab3b7e2264b1b6b1ab2b1b3Aa","unstructured":"Bangerth W. and Rannacher R.,\nFinite element approximation of the acoustic wave equation: Error control and mesh adaptation,\nEast-West J. Numer. Math. 7 (1999), no. 4, 263\u2013282."},{"key":"2023033112444849263_j_cmam-2016-0015_ref_004_w2aab3b7e2264b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"Bangerth W. and Rannacher R.,\nAdaptive Finite Element Methods for Differential Equations,\nBirkh\u00e4user, Basel, 2003.","DOI":"10.1007\/978-3-0348-7605-6"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_005_w2aab3b7e2264b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"Braess D.,\nFinite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd ed.,\nCambridge University Press, Cambridge, 2007.","DOI":"10.1017\/CBO9780511618635"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_006_w2aab3b7e2264b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"Demkowicz L. F. and Gopalakrishnan J.,\nAn overview of the discontinuous Petrov\u2013Galerkin method,\nRecent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations,\nIMA Vol. Math. Appl.,\nSpringer, Cham (2014), 149\u2013180.","DOI":"10.1007\/978-3-319-01818-8_6"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_007_w2aab3b7e2264b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"Dumbser M., K\u00e4ser M. and Toro E. F.,\nAn arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes \u2013 V. Local time stepping and p-adaptivity,\nGeophys. J. Int. 171 (2007), 695\u2013717.","DOI":"10.1111\/j.1365-246X.2007.03427.x"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_008_w2aab3b7e2264b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"Egger H., Kretzschmar F., Schnepp S. M. and Weiland T.,\nA space-time discontinuous Galerkin\u2013Trefftz method for time dependent Maxwell\u2019s equations,\nSIAM J. Sci. Comput. 37 (2015), no. 5, 689\u2013711.","DOI":"10.1137\/140999323"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_009_w2aab3b7e2264b1b6b1ab2b1b9Aa","unstructured":"Ellis T. E., Demkowicz L. F., Chan J. L. and Moser R. D.,\nSpace-time DPG: Designing a method for massively parallel CFD,\nICES report 14-32, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 2014."},{"key":"2023033112444849263_j_cmam-2016-0015_ref_010_w2aab3b7e2264b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"Emmett M. and Minion M. L.,\nToward an efficient parallel in time method for partial differential equations,\nCommun. Appl. Math. Comput. Sci. 7 (2012), 105\u2013132.","DOI":"10.2140\/camcos.2012.7.105"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_011_w2aab3b7e2264b1b6b1ab2b1c11Aa","unstructured":"Evans L. C.,\nPartial Differential Equations, 2nd ed.,\nAmerican Mathematical Society, Providence, 2010."},{"key":"2023033112444849263_j_cmam-2016-0015_ref_012_w2aab3b7e2264b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"Falgout R. D., Friedhoff S., Kolev T. V., MacLachlan S. P. and Schroder J. B.,\nParallel time integration with multigrid,\nSIAM J. Sci. Comput. 36 (2014), no. 6, 635\u2013661.","DOI":"10.1137\/130944230"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_013_w2aab3b7e2264b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"Gander M. J.,\n50 years of time parallel time integration,\nMultiple Shooting and Time Domain Decomposition,\nContrib. Math. Comput. Sci.,\nSpringer, Basel (2015), 69\u2013113.","DOI":"10.1007\/978-3-319-23321-5_3"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_014_w2aab3b7e2264b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"Gander M. J., Halpern L. and Nataf F.,\nOptimal Schwarz waveform relaxation for the one dimensional wave equation,\nSIAM J. Numer. Anal. 41 (2003), no. 5, 1643\u20131681.","DOI":"10.1137\/S003614290139559X"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_015_w2aab3b7e2264b1b6b1ab2b1c15Aa","unstructured":"Gander M. J. and Neum\u00fcller M.,\nAnalysis of a new space-time parallel multigrid algorithm for parabolic problems,\npreprint 2014, http:\/\/arxiv.org\/abs\/1411.0519."},{"key":"2023033112444849263_j_cmam-2016-0015_ref_016_w2aab3b7e2264b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"Gander M. J. and Vandewalle S.,\nAnalysis of the parareal time-parallel time-integration method,\nSIAM J. Sci. Comput. 29 (2007), no. 2, 556\u2013578.","DOI":"10.1137\/05064607X"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_017_w2aab3b7e2264b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"Grote M. J. and Sch\u00f6tzau D.,\nOptimal error estimates for the fully discrete interior penalty DG method for the wave equation,\nJ. Sci. Comput. 40 (2009), no. 1\u20133, 257\u2013272.","DOI":"10.1007\/s10915-008-9247-z"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_018_w2aab3b7e2264b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"Hesthaven J. S. and Warburton T.,\nNodal Discontinuous Galerkin Methods,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-72067-8"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_019_w2aab3b7e2264b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"Heuveline V. and Rannacher R.,\nDuality-based adaptivity in the hp-finite element method,\nJ. Numer. Math. 11 (2003), 95\u2013113.","DOI":"10.1515\/156939503766614126"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_020_w2aab3b7e2264b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"Hochbruck M., Pazur T., Schulz A., Thawinan E. and Wieners C.,\nEfficient time integration for discontinuous Galerkin approximations of linear wave equations,\nZAMM Z. Angew. Math. Mech. 95 (2015), no. 3, 237\u2013259.","DOI":"10.1002\/zamm.201300306"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_021_w2aab3b7e2264b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"Houston P. and S\u00fcli E.,\nhp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems,\nSIAM J. Sci. Comput. 23 (2006), no. 4, 1226\u20131252.","DOI":"10.1137\/S1064827500378799"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_022_w2aab3b7e2264b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"K\u00f6cher U. and Bause M.,\nVariational space-time methods for the wave equation,\nJ. Sci. Comput. 61 (2014), no. 2, 424\u2013453.","DOI":"10.1007\/s10915-014-9831-3"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_023_w2aab3b7e2264b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"Kretzschmar F., Moiola A., Perugia I. and Schnepp S. M.,\nA priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems,\nIMA J. Numer. Anal. (2015), 10.1093\/imanum\/drv064.","DOI":"10.1093\/imanum\/drv064"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_024_w2aab3b7e2264b1b6b1ab2b1c24Aa","doi-asserted-by":"crossref","unstructured":"Lions J.-L., Maday Y. and Turinici G.,\nA parareal in time discretization of PDE\u2019s,\nC. R. Acad. Sci. Paris Ser. I 332 (2001), no. 7, 661\u2013668.","DOI":"10.1016\/S0764-4442(00)01793-6"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_025_w2aab3b7e2264b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"Maurer D. and Wieners C.,\nA parallel block LU decomposition method for distributed finite element matrices,\nParallel Comput. 37 (2011), no. 12, 742\u2013758.","DOI":"10.1016\/j.parco.2011.05.007"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_026_w2aab3b7e2264b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"Nguyen N. C., Peraire J. and Cockburn B.,\nHigh-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics,\nJ. Comput. Phys. 230 (2011), no. 10, 3695\u20133718.","DOI":"10.1016\/j.jcp.2011.01.035"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_027_w2aab3b7e2264b1b6b1ab2b1c27Aa","doi-asserted-by":"crossref","unstructured":"Oden J. T., Prudhomme S. and Demkowicz L.,\nA posteriori error estimation for acoustic wave propagation problems,\nArch. Comput. Methods Eng. 12 (2005), no. 4, 343\u2013389.","DOI":"10.1007\/BF02736190"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_028_w2aab3b7e2264b1b6b1ab2b1c28Aa","doi-asserted-by":"crossref","unstructured":"van der Vegt J. J. W. and Rhebergen S.,\nhp-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I: Multilevel analysis,\nJ. Comput. Phys. 231 (2012), no. 22, 7537\u20137563.","DOI":"10.1016\/j.jcp.2012.05.038"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_029_w2aab3b7e2264b1b6b1ab2b1c29Aa","doi-asserted-by":"crossref","unstructured":"Wang D., Tezaur R. and Farhat C.,\nA hybrid discontinuous in space and time Galerkin method for wave propagation problems,\nInternat. J. Numer. Methods Engrg. 99 (2014), no. 4, 263\u2013289.","DOI":"10.1002\/nme.4673"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_030_w2aab3b7e2264b1b6b1ab2b1c30Aa","doi-asserted-by":"crossref","unstructured":"Wieners C.,\nA geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing,\nComput. Vis. Sci. 13 (2010), 161\u2013175.","DOI":"10.1007\/s00791-010-0135-3"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_031_w2aab3b7e2264b1b6b1ab2b1c31Aa","doi-asserted-by":"crossref","unstructured":"Wieners C. and Wohlmuth B.,\nRobust operator estimates and the application to substructuring methods for first-order systems,\nESAIM Math. Model. Numer. Anal. 48 (2014), 161\u2013175.","DOI":"10.1051\/m2an\/2014006"},{"key":"2023033112444849263_j_cmam-2016-0015_ref_032_w2aab3b7e2264b1b6b1ab2b1c32Aa","doi-asserted-by":"crossref","unstructured":"Zitelli J., Muga I., Demkowicz L., Gopalakrishnan J., Pardo D. and Calo V. M.,\nA class of discontinuous Petrov\u2013Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D,\nJ. Comput. Phys. 230 (2011), no. 7, 2406\u20132432.","DOI":"10.1016\/j.jcp.2010.12.001"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/16\/3\/article-p409.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0015\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0015\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:10:21Z","timestamp":1680282621000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0015\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,7]]},"references-count":32,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,3,10]]},"published-print":{"date-parts":[[2016,7,1]]}},"alternative-id":["10.1515\/cmam-2016-0015"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2016-0015","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2016,4,7]]}}}