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A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

Please always quote using this URN: urn:nbn:de:0297-zib-10332
  • 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.

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Metadaten
Author:Xavier Antoine, Anton Arnold, Christophe Besse, Matthias Ehrhardt, Achim Schädle
Document Type:ZIB-Report
Tag:Schrödinger equation; discrete convolution; transparent boundary condition; unbounded domain
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q40 PDEs in connection with quantum mechanics
45-XX INTEGRAL EQUATIONS / 45Kxx Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] / 45K05 Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M12 Stability and convergence of numerical methods
PACS-Classification:00.00.00 GENERAL / 02.00.00 Mathematical methods in physics / 02.70.-c Computational techniques; simulations (for quantum computation, see 03.67.Lx; for computational techniques extensively used in subdivisions of physics, see the appropriate section; for example, see 47.11.-j Computational methods in fluid dynamics) / 02.70.Bf Finite-difference methods
Date of first Publication:2007/11/09
Series (Serial Number):ZIB-Report (07-34)
ISSN:1438-0064
ZIB-Reportnumber:07-34
Published in:Appeared in: Commun. Comput. Phys. 4 (2008), 729-796

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