Volume 11, Issue 2
A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation

Hisham bin Zubair, Bram Reps & Wim Vanroose

Commun. Comput. Phys., 11 (2012), pp. 415-434.

Published online: 2012-12

Preview Full PDF 136 998
Export citation
  • Abstract

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.


  • Keywords

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-11-415, author = {}, title = {A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation}, journal = {Communications in Computational Physics}, year = {2012}, volume = {11}, number = {2}, pages = {415--434}, abstract = {

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.


}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.121209.180910s}, url = {http://global-sci.org/intro/article_detail/cicp/7370.html} }
TY - JOUR T1 - A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation JO - Communications in Computational Physics VL - 2 SP - 415 EP - 434 PY - 2012 DA - 2012/12 SN - 11 DO - http://dor.org/10.4208/cicp.121209.180910s UR - https://global-sci.org/intro/article_detail/cicp/7370.html KW - AB -

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.


Hisham bin Zubair, Bram Reps & Wim Vanroose. (2020). A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation. Communications in Computational Physics. 11 (2). 415-434. doi:10.4208/cicp.121209.180910s
Copy to clipboard
The citation has been copied to your clipboard