Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-10T03:20:46.654Z Has data issue: false hasContentIssue false
  • English
  • Français

An Explicit Hermite-Taylor Method for the Schrödinger Equation

Part of: Numerical analysis: Ordinary differential equations Applications to specific physical systems

Published online by Cambridge University Press:  27 March 2017

Daniel Appelö ,
Gunilla Kreiss  and
Siyang Wang
Daniel Appelö*
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, 1 University of New Mexico, Albuquerque, NM 87131, USA
Gunilla Kreiss*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden
Siyang Wang*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden
*
*Corresponding author. Email addresses:appelo@math.unm.edu (D. Appelö), gunilla.kreiss@it.uu.se (G. Kreiss), siyang.wang@it.uu.se (S.Wang)
*Corresponding author. Email addresses:appelo@math.unm.edu (D. Appelö), gunilla.kreiss@it.uu.se (G. Kreiss), siyang.wang@it.uu.se (S.Wang)
*Corresponding author. Email addresses:appelo@math.unm.edu (D. Appelö), gunilla.kreiss@it.uu.se (G. Kreiss), siyang.wang@it.uu.se (S.Wang)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

An explicit spectrally accurate order-adaptive Hermite-Taylor method for the Schrödinger equation is developed. Numerical experiments illustrating the properties of the method are presented. The method, which is able to use very coarse grids while still retaining high accuracy, compares favorably to an existing exponential integrator – high order summation-by-parts finite difference method.

Keywords

Hermite methodSchrödinger equationhigh order

MSC classification

Secondary: 65L20: Stability and convergence of numerical methods 81V10: Electromagnetic interaction; quantum electrodynamics

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 5 , May 2017 , pp. 1207 - 1230
Copyright
Copyright © Global-Science Press 2017 

Footnotes

Communicated by Chi-Wang Shu

References

[1] Appelö, D. and Hagstrom, T.. On advection by Hermite methods. Paciffic Journal Of Applied Mathematics, 4(2):125–139, 2011.Google Scholar
[2] Carpenter, Mark H., Gottlieb, David, and Abarbanel, Saul. Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys., 111(2):220–236, 1994.Google Scholar
[3] Chen, R. and Hagstrom, T.. p-adaptive Hermite methods for initial value problems. ESAIM: Mathematical Modelling and Numerical Analysis, 46:545–557, 2012.CrossRefGoogle Scholar
[4] Chen, X., Appelö, D., and Hagstrom, T.. A hybrid Hermite–discontinuous Galerkin method for hyperbolic systems with application to Maxwell's equations. Journal of Computational Physics, 257, Part A:501–520, 2014.Google Scholar
[5] Goodrich, J., Hagstrom, T., and Lorenz, J.. Hermite methods for hyperbolic initial-boundary value problems. Math. Comp., 75:595–630, 2006.Google Scholar
[6] Hagstrom, T. and Appelö, D.. Solving PDEs with Hermite Interpolation. In Springer Lecture Notes in Computational Science and Engineering, 2015.Google Scholar
[7] Hochbruck, M., Lubich, C., and Selhofer, H.. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19:1552–1574, 1998.Google Scholar
[8] Hochbruck, M. and Ostermann, A.. Exponential integrators. Acta Numerica, 19:209–286, 2010.Google Scholar
[9] Kormann, K. and Kronbichler, M.. Parallel finite element operator application: graph partitioning and coloring. 2011 Seventh IEEE International Conference on eScience, pages 332–339, 2011.Google Scholar
[10] Kormann, K. and Larsson, E.. A Galerkin radial basis function method for the Schrödinger equation. SIAM J. Sci. Comput., 35:A2832–A2855, 2013.Google Scholar
[11] Kreiss, H.-O. and Scherer, G.. Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical Aspects of Finite Element in Partial Differential Equations. Academic Press, Inc., 1974.Google Scholar
[12] Mattsson, K. and Nordström, J.. Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys., 199:503–540, 2004.CrossRefGoogle Scholar
[13] Nissen, Anna, Kreiss, Gunilla, and Gerritsen, Margot. High order stable finite difference methods for the Schrödinger equation. J. Sci. Comput., 55:173–199, 2013.Google Scholar