{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,26]],"date-time":"2026-10-26T18:40:35Z","timestamp":1793040035271,"version":"4.5.0"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T00:00:00Z","timestamp":1605484800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T00:00:00Z","timestamp":1605484800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,12]]},"DOI":"10.1007\/s00211-020-01160-w","type":"journal-article","created":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T22:02:46Z","timestamp":1605564166000},"page":"729-782","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form"],"prefix":"10.1007","volume":"146","author":[{"given":"Kenneth","family":"Duru","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leonhard","family":"Rannabauer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alice-Agnes","family":"Gabriel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gunilla","family":"Kreiss","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Bader","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,11,16]]},"reference":[{"key":"1160_CR1","series-title":"Applied Mathematics and Mechanics","volume-title":"Wave Propagation in Elastic Solids","author":"VV Achenbach","year":"1973","unstructured":"Achenbach, V.V.: Wave Propagation in Elastic Solids. Applied Mathematics and Mechanics, vol. 6. North-Holland, Amsterdam (1973)"},{"key":"1160_CR2","doi-asserted-by":"publisher","first-page":"642","DOI":"10.1016\/j.jcp.2005.11.006","volume":"215","author":"D Appel\u00f6","year":"2006","unstructured":"Appel\u00f6, D., Kreiss, G.: A new absorbing layer for elastic waves. J. Comput. Phys. 215, 642\u2013660 (2006)","journal-title":"J. Comput. Phys."},{"key":"1160_CR3","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.jcp.2005.03.011","volume":"209","author":"NA Petersson","year":"2005","unstructured":"Petersson, N.A., Sj\u00f6green, B.: Perfectly matched layer for Maxwell\u2019s equation in second order formulation. J. Comput. Phys. 209, 19\u201346 (2005)","journal-title":"J. Comput. Phys."},{"key":"1160_CR4","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.cma.2012.05.007","volume":"241\u2013244","author":"D Baffet","year":"2012","unstructured":"Baffet, D., Bielak, J., Givoli, D., Hagstrom, T., Rabinovich, D.: Long-time stable high-order absorbing boundary conditions for elastodynamics. Comput. Methods Appl. Mech. Engrg. 241\u2013244, 20\u201337 (2012)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"1160_CR5","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1016\/S0021-9991(03)00184-0","volume":"188","author":"E B\u00e9cache","year":"2003","unstructured":"B\u00e9cache, E., Fauqueux, S., Joly, P.: Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 399\u2013433 (2003)","journal-title":"J. Comput. Phys."},{"key":"1160_CR6","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1006\/jcph.1994.1159","volume":"114","author":"J-P B\u00e9renger","year":"1994","unstructured":"B\u00e9renger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185\u2013200 (1994)","journal-title":"J. Comput. Phys."},{"key":"1160_CR7","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1002\/mop.4650071304","volume":"7","author":"W Chew","year":"1994","unstructured":"Chew, W., Weedon, W.: A 3-d perfectly matched medium from modified Maxwell\u2019s equations with stretched coordinates. IEEE Microw. Opt. Technol. Lett. 7, 599\u2013604 (1994)","journal-title":"IEEE Microw. Opt. Technol. Lett."},{"key":"1160_CR8","doi-asserted-by":"publisher","first-page":"B10302","DOI":"10.1029\/2008JB006271","volume":"114","author":"J de la Puente","year":"2009","unstructured":"de la Puente, J., Ampuero, J.-P., K\u00e4ser, M.: Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method. J. Geophys. Res. 114, B10302 (2009)","journal-title":"J. Geophys. Res."},{"key":"1160_CR9","doi-asserted-by":"publisher","first-page":"3820","DOI":"10.1016\/j.cma.2005.02.031","volume":"195","author":"J Diaz","year":"2006","unstructured":"Diaz, J., Joly, P.: A time domain analysis of pml models in acoustics. Comput. Methods Appl. Mech. Eng. 195, 3820\u20133853 (2006)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1160_CR10","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1111\/j.1365-246X.2006.03051.x","volume":"166","author":"M Dumbser","year":"2006","unstructured":"Dumbser, M., K\u00e4ser, M.: An arbitrary high-order discontinuous galerkin method for elastic waves on unstructured meshes\u2014 I. The two-dimensional isotropic case with external source terms. Geophys. J. Int. 166, 855\u2013877 (2006)","journal-title":"Geophys. J. Int."},{"key":"1160_CR11","doi-asserted-by":"publisher","first-page":"824","DOI":"10.1016\/j.jcp.2016.02.015","volume":"5","author":"M Dumbser","year":"2016","unstructured":"Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ader schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys 5, 824\u2013862 (2016)","journal-title":"J. Comput. Phys"},{"key":"1160_CR12","unstructured":"Duru, K.: Perfectly matched layers and high order difference methods for wave equations. PhD Thesis, Uppsala University Sweden (2012)"},{"key":"1160_CR13","doi-asserted-by":"publisher","first-page":"A1171","DOI":"10.1137\/140976443","volume":"38","author":"K Duru","year":"2016","unstructured":"Duru, K.: The role of numerical boundary procedures in the stability of perfectly matched layers. SIAM J. Sci. Comput. 38, A1171\u2013A1194 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"1160_CR14","doi-asserted-by":"publisher","first-page":"898","DOI":"10.1016\/j.cma.2019.02.036","volume":"350","author":"K Duru","year":"2019","unstructured":"Duru, K., Gabriel, A.-A., Kreiss, G.: On energy stable discontinuous galerkin spectral element approximations of the perfectly matched layer for the wave equation. Comput. Methods Appl. Mech. Eng. 350, 898\u2013937 (2019)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1160_CR15","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1016\/j.jcp.2015.09.048","volume":"303","author":"K Duru","year":"2015","unstructured":"Duru, K., Kozdon, J.E., Kreiss, G.: Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form. J. Comput. Phys. 303, 372\u2013395 (2015)","journal-title":"J. Comput. Phys."},{"key":"1160_CR16","doi-asserted-by":"publisher","first-page":"2883","DOI":"10.1137\/13093563X","volume":"52","author":"K Duru","year":"2014","unstructured":"Duru, K., Kreiss, G.: Boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form. SIAM Numer. Anal. 52, 2883\u20132904 (2014)","journal-title":"SIAM Numer. Anal."},{"key":"1160_CR17","unstructured":"Duru, K., Rannabauer, L., Gabriel, A.-A., Igel, H.: A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes (2017). arXiv:1802.06380"},{"key":"1160_CR18","unstructured":"Duru, K., Rannabauer, L., Gabriel, A.-A., Ling, O.K.A., Igel, H., Bader, M.: A stable discontinuous galerkin method for linear elastodynamics in geometrically complex media using physics based numerical fluxes (2019). arXiv:1907.02658"},{"issue":"5","key":"1160_CR19","doi-asserted-by":"publisher","first-page":"1765","DOI":"10.1073\/pnas.74.5.1765","volume":"74","author":"B Engquist","year":"1977","unstructured":"Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. USA 74(5), 1765\u20131766 (1977)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"1160_CR20","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.wavemoti.2003.12.004","volume":"39","author":"D Givoli","year":"2004","unstructured":"Givoli, D.: High-order local non-reflecting boundary conditions: a review. Wave Motion 39, 319\u2013326 (2004)","journal-title":"Wave Motion"},{"key":"1160_CR21","volume-title":"Time Dependent Problems and Difference Methods","author":"B Gustafsson","year":"1995","unstructured":"Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)"},{"key":"1160_CR22","volume-title":"Topics in Computational Wave Propagation","author":"T Hagstrom","year":"2003","unstructured":"Hagstrom, T.: New results on absorbing layers and radiation boundary conditions. In: Ainsworth, M., et al. (eds.) Topics in Computational Wave Propagation. Springer, Berlin (2003)"},{"key":"1160_CR23","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1142\/S1793744211000291","volume":"3","author":"L Halpern","year":"2011","unstructured":"Halpern, L., Petit-Bergez, S., Rauch, J.: The analysis of matched layers. Conflu. Math. 3, 159\u2013236 (2011)","journal-title":"Conflu. Math."},{"key":"1160_CR24","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1006\/jcph.2002.7118","volume":"181","author":"JS Hesthaven","year":"2002","unstructured":"Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell\u2019s equations. J. Comput. Phys. 181, 186\u2013221 (2002)","journal-title":"J. Comput. Phys."},{"key":"1160_CR25","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1093\/gji\/ggu298","volume":"199","author":"L M\u00e9tivier","year":"2014","unstructured":"M\u00e9tivier, L., Tago, J., Virieux, J.: Smart layers: a simple and robust alternative to PML approaches for elastodynamics. Geophys. J. Int. 199, 700\u2013706 (2014)","journal-title":"Geophys. J. Int."},{"key":"1160_CR26","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1111\/j.1365-246X.2009.04177.x","volume":"178","author":"M Kristekov\u00e1","year":"2009","unstructured":"Kristekov\u00e1, M., Kristek, J., Moczo, P.: Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals. Geophys. J. Int. 178, 813\u2013825 (2009)","journal-title":"Geophys. J. Int."},{"key":"1160_CR27","doi-asserted-by":"publisher","first-page":"1836","DOI":"10.1785\/0120060012","volume":"96","author":"M Kristekov\u00e1","year":"2006","unstructured":"Kristekov\u00e1, M., Kristek, J., Moczo, P., Day, S.M.: Misfit criteria for quantitative comparison of seismograms. Bull. Seism. Soc. Am. 96, 1836\u20131850 (2006)","journal-title":"Bull. Seism. Soc. Am."},{"key":"1160_CR28","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1109\/75.544545","volume":"6","author":"M Kuzuoglu","year":"1996","unstructured":"Kuzuoglu, M., Mittra, R.: Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic. IEEE Microw. Guided Wave Lett. 6, 447\u2013449 (1996)","journal-title":"IEEE Microw. Guided Wave Lett."},{"key":"1160_CR29","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1061\/JMCEA3.0001144","volume":"95","author":"J Lysmer","year":"1969","unstructured":"Lysmer, J., Kuhlemeyer, R.L.: Finite dynamic model for infinite media. J. Eng. Mech. Div. ASCE 95, 859\u2013877 (1969)","journal-title":"J. Eng. Mech. Div. ASCE"},{"key":"1160_CR30","volume-title":"Mathematical Foundations of Elasticity","author":"JE Marsden","year":"1994","unstructured":"Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover Publications Inc., New York (1994)"},{"key":"1160_CR31","doi-asserted-by":"publisher","first-page":"684","DOI":"10.1016\/j.camwa.2016.12.027","volume":"73","author":"A Modave","year":"2017","unstructured":"Modave, A., Lambrechts, J., Geuzaine, C.: Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations. Comput. Math Appl. 73, 684\u2013700 (2017)","journal-title":"Comput. Math Appl."},{"key":"1160_CR32","doi-asserted-by":"crossref","first-page":"B02309","DOI":"10.1029\/2011JB008857","volume":"117","author":"C Pelties","year":"2012","unstructured":"Pelties, C., de la Puente, J., Ampuero, J.-P., Brietzke, G.B., K\u00e4ser, M.: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. 117, B02309 (2012)","journal-title":"J. Geophys. Res."},{"key":"1160_CR33","doi-asserted-by":"publisher","first-page":"532","DOI":"10.1016\/j.jcp.2016.05.060","volume":"321","author":"NA Petersson","year":"2016","unstructured":"Petersson, N.A., O\u2019Reilly, O., Sj\u00f6green, B., Bydlon, S.: Discretizing singular point sources in hyperbolic wave propagation problems. J. Comput. Phys. 321, 532\u2013555 (2016)","journal-title":"J. Comput. Phys."},{"key":"1160_CR34","doi-asserted-by":"publisher","first-page":"107251","DOI":"10.1016\/j.cpc.2020.107251","volume":"254","author":"A Reinarz","year":"2020","unstructured":"Reinarz, A., Charrier, D.E., Bader, M., Bovard, L., Dumbser, M., Fambri, F., Duru, K., Gabriel, A.-A., Gallard, J.-M., K\u00f6ppel, S., Krenz, L., Rannabauer, L., Rezzolla, L., Samfass, P., Tavelli, M., Weinzierl, T.: Exahype: an engine for parallel dynamically adaptive simulations of wave problems. Comput. Phys. Comm. 254, 107251 (2020)","journal-title":"Comput. Phys. Comm."},{"key":"1160_CR35","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1002\/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A","volume":"27","author":"JA Roden","year":"2000","unstructured":"Roden, J.A., Gedney, S.D.: Convolution PML (CPML): an efficient fdtd implementation of the CFS-PML for arbitrary media. IEEE Microw. Opt. Technol. Lett. 27, 334\u2013339 (2000)","journal-title":"IEEE Microw. Opt. Technol. Lett."},{"key":"1160_CR36","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1016\/j.wavemoti.2007.03.001","volume":"44","author":"EA Skelton","year":"2007","unstructured":"Skelton, E.A., Adams, S.D.M., Craster, R.V.: Guided elastic waves and perfectly matched layers. Wave Motion 44, 573\u2013592 (2007)","journal-title":"Wave Motion"},{"key":"1160_CR37","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1109\/TAP.2017.2769132","volume":"66","author":"Q Sun","year":"2018","unstructured":"Sun, Q., Zhang, R., Zhan, Q., Liu, Q.H.: A novel coupling algorithm for perfectly matched layer with wave equation-based discontinuous galerkin time-domain method. IEEE Trans. Antennas Propag. 66, 255\u2013261 (2018)","journal-title":"IEEE Trans. Antennas Propag."},{"key":"1160_CR38","doi-asserted-by":"publisher","unstructured":"Toro, E.F.: The equations of fluid dynamics. In: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin, Heidelberg (2009). https:\/\/doi.org\/10.1007\/b79761_1","DOI":"10.1007\/b79761_1"},{"key":"1160_CR39","doi-asserted-by":"crossref","unstructured":"Uphoff, C., Rettenberger, S., Bader, M., Madden, E.H., Ulrich, T., Wollherr, S., Gabriel, A.-A.: Extreme scale multi-physics simulations of the tsunamigenic 2004 sumatra megathrust earthquake. In: Proceeding SC \u201917 Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2018)","DOI":"10.1145\/3126908.3126948"},{"key":"1160_CR40","doi-asserted-by":"publisher","first-page":"1714","DOI":"10.1093\/gji\/ggu219","volume":"198","author":"Z Xie","year":"2014","unstructured":"Xie, Z., Martin, R., Komatitsch, D., Matzen, R.: Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element pml. Geophys. J. Int. 198, 1714\u20131747 (2014)","journal-title":"Geophys. J. Int."},{"key":"1160_CR41","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1190\/1.3560019","volume":"76","author":"C Zeng","year":"2011","unstructured":"Zeng, C., Xia, J., Miller, R., Tsoflias, G.: Application of the multi-axial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves. Geophysics 76, 43\u201352 (2011)","journal-title":"Geophysics"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01160-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-020-01160-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01160-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,14]],"date-time":"2021-04-14T13:25:39Z","timestamp":1618406739000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-020-01160-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,16]]},"references-count":41,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["1160"],"URL":"https:\/\/doi.org\/10.1007\/s00211-020-01160-w","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,16]]},"assertion":[{"value":"26 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 September 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 October 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}