{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T00:01:16Z","timestamp":1770595276174,"version":"3.49.0"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,12,24]],"date-time":"2016-12-24T00:00:00Z","timestamp":1482537600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11272009"],"award-info":[{"award-number":["11272009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11502208"],"award-info":[{"award-number":["11502208"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11521202"],"award-info":[{"award-number":["11521202"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2017,7]]},"DOI":"10.1007\/s10915-016-0344-0","type":"journal-article","created":{"date-parts":[[2016,12,24]],"date-time":"2016-12-24T19:43:54Z","timestamp":1482608634000},"page":"1-13","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Exact Boundary Condition for Semi-discretized Schr\u00f6dinger Equation and Heat Equation in a Rectangular Domain"],"prefix":"10.1007","volume":"72","author":[{"given":"Gang","family":"Pang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yibo","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaoqiang","family":"Tang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,12,24]]},"reference":[{"key":"344_CR1","unstructured":"Greengard, L., Lin, P.: On the Numerical Solution of the Heat Equation in Unbounded Domains (Part I). Tech. Note 98002, Courant Mathematics and Computing Laboratory, New York University (1998)"},{"key":"344_CR2","doi-asserted-by":"crossref","first-page":"74","DOI":"10.4208\/jcm.1004-m3195","volume":"29","author":"X Wu","year":"2011","unstructured":"Wu, X., Zhang, J.: High-order local absorbing boundary conditions for heat equation in unbounded domains. J. Comput. Math 29, 74 (2011)","journal-title":"J. Comput. Math"},{"issue":"1","key":"344_CR3","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1137\/S1064827594277053","volume":"21","author":"T Fevens","year":"1999","unstructured":"Fevens, T., Jiang, H.: Absorbing boundary conditions for the Schr\u00f6dinger equation. SIAM J. Sci. Comput 21(1), 255 (1999)","journal-title":"SIAM J. Sci. Comput"},{"key":"344_CR4","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.apnum.2004.01.001","volume":"50","author":"X Wu","year":"2004","unstructured":"Wu, X., Sun, Z.: Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions. Appl. Numer. Math 50, 261 (2004)","journal-title":"Appl. Numer. Math"},{"issue":"3","key":"344_CR5","doi-asserted-by":"crossref","first-page":"501","DOI":"10.4310\/CMS.2003.v1.n3.a7","volume":"1","author":"A Arnold","year":"2003","unstructured":"Arnold, A., Ehrhardt, M., Sofronov, I.: Approximation, stability and fast calculation of non-local boundary conditions for the Schr\u00f6dinger equation. Commun. Math. Sci 1(3), 501 (2003)","journal-title":"Commun. Math. Sci"},{"key":"344_CR6","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.wavemoti.2003.12.007","volume":"39","author":"T Hagstrom","year":"2004","unstructured":"Hagstrom, T., Warburton, T.: A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems. Wave Motion 39, 327 (2004)","journal-title":"Wave Motion"},{"key":"344_CR7","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1137\/S0036142998336916","volume":"37","author":"B Alpert","year":"2000","unstructured":"Alpert, B., Greengard, L., Hagstrom, T.: Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal 37, 1138 (2000)","journal-title":"SIAM J. Numer. Anal"},{"key":"344_CR8","first-page":"127","volume":"52","author":"M Grote","year":"1996","unstructured":"Grote, M., Keller, J.: Nonreflecting boundary conditions for time dependent scattering. J. Comput. Phys. 52, 127 (1996)","journal-title":"J. Comput. Phys."},{"key":"344_CR9","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0165-2125(91)90053-Q","volume":"14","author":"V Baskakov","year":"1991","unstructured":"Baskakov, V., Popov, A.: Implementation of transparent boundaries for numerical solution of the Schr\u00f6dinger equation. Wave Motion 14, 123 (1991)","journal-title":"Wave Motion"},{"issue":"1","key":"344_CR10","doi-asserted-by":"crossref","first-page":"79","DOI":"10.4310\/CMS.2004.v2.n1.a5","volume":"2","author":"H Han","year":"2004","unstructured":"Han, H., Huang, Z.: Exact artificial boundary conditions for the Schr\u00f6dinger equation in $$\\mathbb{R}^2$$ R 2 . Commun. Math. Sci. 2(1), 79 (2004)","journal-title":"Commun. Math. Sci."},{"issue":"3","key":"344_CR11","doi-asserted-by":"crossref","first-page":"889","DOI":"10.4310\/CMS.2012.v10.n3.a9","volume":"10","author":"A Arnold","year":"2012","unstructured":"Arnold, A., Ehrhardt, M., Schulte, M., Sofronov, I.: Discrete transparent boundary conditions for the Schr\u00f6dinger equation on circular domains. Commun. Math. Sci 10(3), 889 (2012)","journal-title":"Commun. Math. Sci"},{"key":"344_CR12","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1002\/cpa.20200","volume":"61","author":"S Jiang","year":"2008","unstructured":"Jiang, S., Greengard, L.: Nonreflecting boundary conditions for the time-dependent Schr\u00f6dinger equation in two dimensions. Commun. Pure. Appl. Math 61, 261 (2008)","journal-title":"Commun. Pure. Appl. Math"},{"key":"344_CR13","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1090\/S0025-5718-04-01631-X","volume":"73","author":"X Antoine","year":"2004","unstructured":"Antoine, X., Besse, C., Mouysset, V.: Numerical schemes for the simulation of the two-dimensional Schr\u00f6dinger equation using non-reflecting boundary conditions. Math. Comput. 73, 1779 (2004)","journal-title":"Math. Comput."},{"issue":"2","key":"344_CR14","doi-asserted-by":"crossref","first-page":"1008","DOI":"10.1137\/090780535","volume":"33","author":"X Antoine","year":"2011","unstructured":"Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear Schr\u00f6dinger equations. SIAM J. Sci. Comput. 33(2), 1008 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"344_CR15","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s00211-013-0542-8","volume":"125","author":"X Antoine","year":"2013","unstructured":"Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for the two-dimensional Schr\u00f6dinger equation with an exterior potential. Part II: discretization and numerical results. Numer. Math. 125(2), 191 (2013)","journal-title":"Numer. Math."},{"key":"344_CR16","doi-asserted-by":"crossref","first-page":"1606","DOI":"10.1016\/j.cpc.2014.03.001","volume":"185","author":"H Li","year":"2014","unstructured":"Li, H., Wu, X., Zhang, J.: Local artificial boundary conditions for Schr\u00f6dinger and heat equations by using high-order azimuth derivatives on circular artificial boundary. Comput. Phys. Commun. 185, 1606 (2014)","journal-title":"Comput. Phys. Commun."},{"key":"344_CR17","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.camwa.2015.12.018","volume":"71","author":"L Bian","year":"2016","unstructured":"Bian, L., Ji, S., Pang, G., Tang, S.: Accurate boundary treatment for transient Schr\u00f6dinger equation under polar coordinates. Comput. Math. Appl. 71, 479 (2016)","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"344_CR18","first-page":"729","volume":"4","author":"X Antoine","year":"2008","unstructured":"Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Sch\u00e4dle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schr\u00f6dinger equations. Commun. Comput. Phys. 4(4), 729 (2008)","journal-title":"Commun. Comput. Phys."},{"key":"344_CR19","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/S0898-1221(02)00180-3","volume":"44","author":"H Han","year":"2002","unstructured":"Han, H., Huang, Z.: Exact and approximating boundary conditions for the parabolic problems on unbounded domains. Comput. Math. Appl. 44, 655 (2002)","journal-title":"Comput. Math. Appl."},{"key":"344_CR20","unstructured":"Du, Q., Han, H., Zhang, J., Zheng, C.: Numerical solution of the two-dimensional nonlocal wave equation on unbounded domain, preprint"},{"key":"344_CR21","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1007\/s00466-011-0615-4","volume":"48","author":"G Pang","year":"2011","unstructured":"Pang, G., Tang, S.: Time history kernel functions for square lattice. Comput. Mech. 48, 699 (2011)","journal-title":"Comput. Mech."},{"key":"344_CR22","doi-asserted-by":"crossref","first-page":"1622","DOI":"10.1063\/1.1665785","volume":"12","author":"S Katsura","year":"1971","unstructured":"Katsura, S., Inawashiro, S.: Lattice Greens functions for the rectangular and the square lattices at arbitrary points. J. Math. Phys. 12, 1622 (1971)","journal-title":"J. Math. Phys."},{"issue":"1","key":"344_CR23","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.jcp.2005.08.001","volume":"213","author":"S Tang","year":"2006","unstructured":"Tang, S., Hou, T., Liu, W.: A pseudo-spectral multiscale method: interfacial conditions and coarse grid equations. J. Comput. Phys. 213(1), 57\u201385 (2006)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-016-0344-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-016-0344-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-016-0344-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,16]],"date-time":"2019-09-16T19:03:40Z","timestamp":1568660620000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-016-0344-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,24]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,7]]}},"alternative-id":["344"],"URL":"https:\/\/doi.org\/10.1007\/s10915-016-0344-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,24]]}}}