Advertisement

Stable Perfectly Matched Layers for the Schrödinger Equations

  • Kenneth DuruEmail author
  • Gunilla Kreiss
Conference paper

Abstract

We present a well-posed and stable perfectly matched layer (PML) for the time-dependent Schrödinger wave equations. The layer consists of the Hamiltonian (H) perturbed by a complex absorbing potential (CAP, − iσ1), H cap  = H − iσ1, and carefully chosen auxiliary functions to ensure a zero reflection coefficient at the interface between the physical domain and the layer. Using standard perturbation techniques, we show that the layer is asymptotically stable. Numerical experiments are presented showing the accuracy of the new PML model. The numerical scheme couples the standard Crank–Nicolson scheme for the modified wave equation to an explicit scheme of the Runge–Kutta type for the auxiliary differential equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antoine X., Arnold A., Besse C., Ehrhardt M., Schädle A. A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Non-linear Schrödinger Equations. Commun. Comput. Phys. 4(4) (2008) 729–796MathSciNetGoogle Scholar
  2. 2.
    Berenger J. P. A perfectly Matched Layer for the Absorption of Electromagnetic Waves. J. Comput. Phys. 114 (1994) 185–200zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chew W. C., Weedon W. H. A 3-D Perfectly Matched Medium from Modified Maxwell’s Equations with Stretched Coordinates. Micro. Opt. Tech. Lett. 7(13) (1994) 599–604CrossRefGoogle Scholar
  4. 4.
    Hagstrom T. New results on absorbing layers and radiation boundary conditions, Topics in computational wave propagation, 142, Lect. Notes Comput. Sci. Eng. 31, Springer, Berlin, 2003Google Scholar
  5. 5.
    Santra R. Why Complex Absorbing Potentials Work: A Discrete-Variable-Representation Perspective. Phys. Rev. A 74 (2006) 034701Google Scholar
  6. 6.
    Sjögreen B., Petersson N. A. Perfectly Matched Layer for Maxwell’s Equations in Second Order Formulation. J. Comput. Phys. 209 (2005) 19–46zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Taflove A. Advances in Computational Electrodynamics, The Finite-Difference Time-Domain. Artec House Inc. 1998Google Scholar
  8. 8.
    Zheng C. A Perfectly Matched Layer Approach to the Non-linear Schrodinger Wave Equations. J. Comput. Phys. 227 (2007) 537–556zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Division of Scientific Computing, Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations