{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:00:42Z","timestamp":1776783642157,"version":"3.51.2"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"248","license":[{"start":{"date-parts":[[2005,1,23]],"date-time":"2005-01-23T00:00:00Z","timestamp":1106438400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schr\u00f6dinger equation in a bounded domain\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with artificial boundary conditions set on the arbitrarily shaped boundary of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.\n                  <\/p>","DOI":"10.1090\/s0025-5718-04-01631-x","type":"journal-article","created":{"date-parts":[[2004,6,11]],"date-time":"2004-06-11T15:05:00Z","timestamp":1086966300000},"page":"1779-1799","source":"Crossref","is-referenced-by-count":77,"title":["Numerical schemes for the simulation of the two-dimensional Schr\u00f6dinger equation using non-reflecting boundary conditions"],"prefix":"10.1090","volume":"73","author":[{"given":"Xavier","family":"Antoine","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Besse","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vincent","family":"Mouysset","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2004,1,23]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1137\/S0036142900374433","article-title":"Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schr\u00f6dinger-type equations","volume":"40","author":"Alonso-Mallo, Isa\u00edas","year":"2002","journal-title":"SIAM J. 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