{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,26]],"date-time":"2026-04-26T05:13:37Z","timestamp":1777180417230,"version":"3.51.4"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"334","license":[{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A second-order Crank-Nicolson finite difference method is designed to solve a 1D nonlocal Schr\u00f6dinger equation on the whole real axis. We employ an asymptotically compatible scheme to discretize the spatially nonlocal operator, and apply the Crank-Nicolson scheme in time to achieve a fully discrete infinite system. An iterative technique for the second-order matrix difference equation is then developed to obtain Dirichlet-to-Dirichlet (DtD)-type artificial boundary conditions (ABCs) with the application of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z\">\n                        <mml:semantics>\n                          <mml:mi>z<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -transform for the resulting fully discrete system. After that, with the aid of discrete nonlocal Green\u2019s first identity, we derive Dirichlet-to-Neumann (DtN)-type ABCs from DtD-type ABCs. The resulting DtN-type ABCs are available to reduce the infinite discrete system to a finite discrete system on a truncated computational domain, and make it possible to perform stability and convergence analysis for the reduced problem. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.\n                  <\/p>","DOI":"10.1090\/mcom\/3685","type":"journal-article","created":{"date-parts":[[2021,9,15]],"date-time":"2021-09-15T09:56:53Z","timestamp":1631699813000},"page":"761-783","source":"Crossref","is-referenced-by-count":10,"title":["Stability and error analysis for a second-order approximation of 1D nonlocal Schr\u00f6dinger equation under DtN-type boundary conditions"],"prefix":"10.1090","volume":"91","author":[{"given":"Jihong","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiwei","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunxiong","family":"Zheng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2021,9,28]]},"reference":[{"issue":"4","key":"1","first-page":"729","article-title":"A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schr\u00f6dinger equations","volume":"4","author":"Antoine, Xavier","year":"2008","journal-title":"Commun. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1815-2406","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/S0021-9991(03)00159-1","article-title":"Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schr\u00f6dinger equation","volume":"188","author":"Antoine, X.","year":"2003","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"Anton Arnold. Numerically absorbing boundary conditions for quantum evolution equations. 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