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A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation

Published online by Cambridge University Press:  20 August 2015

Hisham bin Zubair ,
Bram Reps  and
Wim Vanroose
Hisham bin Zubair*
Affiliation:
Department of Mathematical Sciences, Faculty of Computer Science, Institute of Business Administration, University Rd., 75270 Karachi, Pakistan
Bram Reps*
Affiliation:
Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium
Wim Vanroose*
Affiliation:
Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium
*
Corresponding author.Email:hbinzubair@iba.edu.pk
Email address:bram.reps@ua.ac.be
Email address:wim.vanroose@ua.ac.be
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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

65F0865F1065F1565M55ScatteringSchrödinger equationexterior complex scalingpreconditioningmulti-gridcomplex-shifted Laplacian (CSL)complex-scaled grid (CSG)quadrant-definite(QD)
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 415 - 434
Copyright
Copyright © Global Science Press Limited 2012

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