{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T02:32:03Z","timestamp":1768703523251,"version":"3.49.0"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,9,24]],"date-time":"2018-09-24T00:00:00Z","timestamp":1537747200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,9,24]],"date-time":"2018-09-24T00:00:00Z","timestamp":1537747200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-1416742"],"award-info":[{"award-number":["DMS-1416742"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,9]]},"DOI":"10.1007\/s11075-018-0602-3","type":"journal-article","created":{"date-parts":[[2018,9,24]],"date-time":"2018-09-24T06:46:51Z","timestamp":1537771611000},"page":"223-243","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["A new FDTD scheme for Maxwell\u2019s equations in Kerr-type nonlinear media"],"prefix":"10.1007","volume":"82","author":[{"given":"Hongen","family":"Jia","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8338-8412","authenticated-orcid":false,"given":"Jichun","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiwei","family":"Fang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ming","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,9,24]]},"reference":[{"key":"602_CR1","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. Computat. Phys. 114, 185\u2013200 (1994)","journal-title":"J. Computat. Phys."},{"key":"602_CR2","doi-asserted-by":"crossref","unstructured":"Bloembergen, N.: Nonlinear optics world scientific (1996)","DOI":"10.1142\/3046"},{"key":"602_CR3","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1016\/j.jcp.2017.08.009","volume":"350","author":"VA Bokil","year":"2017","unstructured":"Bokil, V.A., Cheng, Y., Jiang, Y., Li, F.: Energy stable discontinuous Galerkin methods for Maxwell\u2019s equations in nonlinear optical media. J. Comput. Phys. 350, 420\u2013452 (2017)","journal-title":"J. Comput. Phys."},{"key":"602_CR4","doi-asserted-by":"publisher","first-page":"1542","DOI":"10.1137\/S0036142998349977","volume":"37","author":"Z Chen","year":"2000","unstructured":"Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37, 1542\u20131570 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"602_CR5","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s002110050417","volume":"82","author":"PJr Ciarlet","year":"1999","unstructured":"Ciarlet, P. Jr, Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell\u2019s equations. Numer. Math. 82, 193\u2013219 (1999)","journal-title":"Numer. Math."},{"issue":"8","key":"602_CR6","doi-asserted-by":"publisher","first-page":"1705","DOI":"10.1007\/s11425-013-4609-x","volume":"56","author":"L Gao","year":"2013","unstructured":"Gao, L., Zhang, B.: Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell\u2019s equations. Sci. China Math. 56(8), 1705\u20131726 (2013)","journal-title":"Sci. China Math."},{"issue":"3","key":"602_CR7","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1142\/S0219891608001623","volume":"5","author":"A Fathy","year":"2008","unstructured":"Fathy, A., Wang, C., Wilson, J., Yang, S.: A fourth order difference scheme for the Maxwell equations on Yee grid. J. Hyperbolic Differ. Equ. 5(3), 613\u2013642 (2008)","journal-title":"J. Hyperbolic Differ. Equ."},{"key":"602_CR8","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1016\/j.jcp.2007.01.031","volume":"225","author":"A Fisher","year":"2007","unstructured":"Fisher, A., White, D., Rodrigue, G.: An efficient vector finite element method for nonlinear electromagnetic modeling. J. Comput. Phys. 225, 1331\u20131346 (2007)","journal-title":"J. Comput. Phys."},{"issue":"11","key":"602_CR9","doi-asserted-by":"publisher","first-page":"444","DOI":"10.1109\/LMWC.2002.805539","volume":"12","author":"M Fujii","year":"2002","unstructured":"Fujii, M., Russer, P.: A nonlinear and dispersive APML ABC for the FD-TD methods. IEEE Microwave Wireless Compon. Lett. 12(11), 444\u2013446 (2002)","journal-title":"IEEE Microwave Wireless Compon. Lett."},{"key":"602_CR10","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1109\/JQE.2003.821881","volume":"40","author":"M Fujii","year":"2004","unstructured":"Fujii, M., Tahara, M., Sakagami, I., Freude, W., Russer, P.: High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media. IEEE J. Quantum Electron. 40, 175\u2013182 (2004)","journal-title":"IEEE J. Quantum Electron."},{"issue":"12","key":"602_CR11","doi-asserted-by":"publisher","first-page":"1630","DOI":"10.1109\/8.546249","volume":"44","author":"SD Gedney","year":"1996","unstructured":"Gedney, S. D.: An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans. Antennas Propag. 44(12), 1630\u20131639 (1996)","journal-title":"IEEE Trans. Antennas Propag."},{"key":"602_CR12","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1142\/S0218202517500099","volume":"27","author":"J Huang","year":"2017","unstructured":"Huang, J., Shu, C.-W.: A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr-Debye model. Math. Models Methods Appl. Sci. 27, 549\u2013580 (2017)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"10","key":"602_CR13","doi-asserted-by":"publisher","first-page":"1251","DOI":"10.1109\/68.329654","volume":"6","author":"RM Joseph","year":"1994","unstructured":"Joseph, R.M., Taflove, A.: Spatial soliton deflection mechanism indicated by FD-TD Maxwell\u2019s equations modeling. IEEE Photon. Technol. Lett. 6(10), 1251\u20131254 (1994)","journal-title":"IEEE Photon. Technol. Lett."},{"key":"602_CR14","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1016\/j.jcp.2013.11.018","volume":"258","author":"J Li","year":"2014","unstructured":"Li, J., Hesthaven, J. S.: Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials. J. Comput. Phys. 258, 915\u2013930 (2014)","journal-title":"J. Comput. Phys."},{"key":"602_CR15","doi-asserted-by":"crossref","unstructured":"Li, J., Huang, Y.: Time-domain Finite Element Methods for Maxwell\u2019s Equations in Metamaterials. Springer Series in Computational Mathematics, vol. 43. Springer (2013)","DOI":"10.1007\/978-3-642-33789-5"},{"issue":"4","key":"602_CR16","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1007\/s00211-015-0788-4","volume":"134","author":"J Li","year":"2016","unstructured":"Li, J., Shields, S.: Superconvergence analysis of Yee scheme for metamaterial Maxwell\u2019s equations on non-uniform rectangular meshes. Numer. Math. 134(4), 741\u2013781 (2016)","journal-title":"Numer. Math."},{"issue":"8","key":"602_CR17","doi-asserted-by":"publisher","first-page":"3476","DOI":"10.1016\/j.jde.2013.01.028","volume":"254","author":"P Li","year":"2013","unstructured":"Li, P., Zheng, W.: An H \u2212 \u03c8 formulation for the three-dimensional eddy current problem in laminated structures. J. Differential Equations 254(8), 3476\u20133500 (2013)","journal-title":"J. Differential Equations"},{"key":"602_CR18","first-page":"775","volume":"10","author":"W Li","year":"2013","unstructured":"Li, W., Liang, D., Lin, Y.: A new energy-conserved s-FDTD scheme for Maxwell\u2019s equations in metamaterials. Int. J. Numer. Anal. Model 10, 775\u2013794 (2013)","journal-title":"Int. J. Numer. Anal. Model"},{"key":"602_CR19","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001","volume-title":"Finite element methods for maxwell\u2019s equations","author":"P Monk","year":"2003","unstructured":"Monk, P.: Finite element methods for maxwell\u2019s equations. Oxford University Press, Oxford (2003)"},{"key":"602_CR20","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1137\/0731021","volume":"32","author":"P Monk","year":"1994","unstructured":"Monk, P., S\u00fcli, E.: A convergence analysis of Yee\u2019s scheme on nonuniform grids. SIAM J. Numer. Anal. 32, 393\u2013412 (1994)","journal-title":"SIAM J. Numer. Anal."},{"key":"602_CR21","unstructured":"Shen, Y.-R.: The Principles of Nonlinear Optics. Wiley Interscience (1994)"},{"key":"602_CR22","doi-asserted-by":"crossref","unstructured":"Taflove, A., Haguess, S.C.: Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd edn. Artech House (2005)","DOI":"10.1016\/B978-012170960-0\/50046-3"},{"key":"602_CR23","doi-asserted-by":"publisher","first-page":"1338","DOI":"10.1137\/17M111314X","volume":"56","author":"H Wu","year":"2018","unstructured":"Wu, H., Zou, J.: Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers. SIAM J. Numer. Anal. 56, 1338\u20131359 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"602_CR24","doi-asserted-by":"publisher","first-page":"2347","DOI":"10.1364\/JOSAA.27.002347","volume":"27","author":"Z Xu","year":"2010","unstructured":"Xu, Z., Bao, G.: A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects. J. Opt. Soc. Am. A 27, 2347\u20132353 (2010)","journal-title":"J. Opt. Soc. Am. A"},{"key":"602_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2017.04.046","volume":"343","author":"L Yuan","year":"2017","unstructured":"Yuan, L., Lu, Y.: Robust iterative method for nonlinear Helmholtz equation. J. Comput. Phys. 343, 1\u20139 (2017)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"602_CR26","doi-asserted-by":"publisher","first-page":"353","DOI":"10.4208\/aamm.2014.m811","volume":"8","author":"Y Zhang","year":"2016","unstructured":"Zhang, Y., Nguyen, D.D., Du, K., Xu, J., Zhao, S.: Time-domain numerical solutions of Maxwell interface problems with discontinuous electromagnetic waves. Adv. Appl. Math. Mech. 8(3), 353\u2013385 (2016)","journal-title":"Adv. Appl. Math. Mech."},{"key":"602_CR27","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1364\/JOSAB.10.000186","volume":"10","author":"RW Ziolkowski","year":"1993","unstructured":"Ziolkowski, R.W., Judkins, J.B.: Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time. J. Optical Soc. Amer. B 10, 186\u2013198 (1993)","journal-title":"J. Optical Soc. Amer. B"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0602-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-018-0602-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0602-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T15:49:49Z","timestamp":1589730589000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-018-0602-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,24]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,9]]}},"alternative-id":["602"],"URL":"https:\/\/doi.org\/10.1007\/s11075-018-0602-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,9,24]]},"assertion":[{"value":"1 February 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 September 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 September 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}