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  <doc>
    <id>1033</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2007-11-09</completedDate>
    <publishedDate>2007-11-09</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A Review of Transparent and Artificial Boundary Conditions Techniques  for Linear and Nonlinear Schrödinger Equations</title>
    <abstract language="eng">In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.</abstract>
    <identifier type="serial">07-34</identifier>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="opus3-id">1061</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-10332</identifier>
    <enrichment key="SourceTitle">Appeared in: Commun. Comput. Phys. 4 (2008), 729-796</enrichment>
    <author>Xavier Antoine</author>
    <submitter>unknown unknown</submitter>
    <author>Anton Arnold</author>
    <author>Christophe Besse</author>
    <author>Matthias Ehrhardt</author>
    <author>Achim Schädle</author>
    <series>
      <title>ZIB-Report</title>
      <number>07-34</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Schrödinger equation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>transparent boundary condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>discrete convolution</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>unbounded domain</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="pacs" number="02.70.Bf">Finite-difference methods</collection>
    <collection role="msc" number="35Q40">PDEs in connection with quantum mechanics</collection>
    <collection role="msc" number="45K05">Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]</collection>
    <collection role="msc" number="65M12">Stability and convergence of numerical methods</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compnano">Computational Nano Optics</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/1033/ZR_07_34.pdf</file>
  </doc>
  <doc>
    <id>955</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2007-05-22</completedDate>
    <publishedDate>2007-05-22</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Transparent boundary conditons for time-dependent problems</title>
    <abstract language="eng">A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.</abstract>
    <identifier type="serial">07-12</identifier>
    <identifier type="opus3-id">955</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-9558</identifier>
    <enrichment key="SourceTitle">Appeared in: SIAM J. Sci. Comput., Vol. 30 (5) , 2358-2385 (2008)</enrichment>
    <author>Daniel Ruprecht</author>
    <author>Achim Schädle</author>
    <author>Frank Schmidt</author>
    <author>Lin Zschiedrich</author>
    <series>
      <title>ZIB-Report</title>
      <number>07-12</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>transparent boundary condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>non-reflecting boundary condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>pole condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>wave equation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Schrödinger equation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>drift diffusion equation</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="30E10">Approximation in the complex domain</collection>
    <collection role="msc" number="65M20">Method of lines</collection>
    <collection role="msc" number="65M60">Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="institutes" number="compnano">Computational Nano Optics</collection>
    <collection role="persons" number="zschiedrich">Zschiedrich, Lin Werner</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/955/ZR-07-12.pdf</file>
    <file>https://opus4.kobv.de/opus4-zib/files/955/ZR-07-12.ps</file>
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