Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T04:53:56.150Z Has data issue: false hasContentIssue false
  • English
  • Français

Plane wave stability of some conservative schemesfor the cubic Schrödinger equation

Published online by Cambridge University Press:  08 July 2009

Morten Dahlby  and
Brynjulf Owren
Morten Dahlby
Affiliation:
Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway. [email protected]; [email protected]
Brynjulf Owren
Affiliation:
Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway. [email protected]; [email protected]
Get access

Abstract

The plane wave stability properties of the conservative schemes of Besse [SIAM J. Numer. Anal.42 (2004) 934–952] and Fei et al. [Appl. Math. Comput.71 (1995) 165–177] for the cubic Schrödinger equation are analysed. Although the two methods possess many of the same conservation properties, we show that their stability behaviour is very different.An energy preserving generalisation of the Fei method with improved stability is presented.

Keywords

Finite difference methodstabilityenergy conservationnonlinear Schrödinger equationlinearly implicit methods.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 677 - 687
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M.J. and Ladik, J.F., A nonlinear difference scheme and inverse scattering. Studies Appl. Math. 55 (1976) 213–229. CrossRef
Berland, H., Owren, B. and Skaflestad, B., Solving the nonlinear Schrödinger equation using exponential integrators. Model. Ident. Control 27 (2006) 201–218. CrossRef
Besse, C., A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42 (2004) 934–952 (electronic). CrossRef
Celledoni, E., Cohen, D. and Owren, B., Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8 (2008) 303–317. CrossRef
Durán, A. and Sanz-Serna, J.M., The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation. IMA J. Numer. Anal. 20 (2000) 235–261. CrossRef
Fei, Z., Pérez-García, V.M. and Vázquez, L., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput. 71 (1995) 165–177.
E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration, Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31. Second Edition, Springer-Verlag, Berlin (2006).
Islas, A.L., Karpeev, D.A. and Schober, C.M., Geometric integrators for the nonlinear Schrödinger equation. J. Comput. Phys. 173 (2001) 116–148. CrossRef
Matsuo, T. and Furihata, D., Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171 (2001) 425–447. CrossRef
Taha, T.R. and Ablowitz, J., Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. J. Comput. Phys. 55 (1984) 203–230. CrossRef
Weideman, J.A.C. and Herbst, B.M., Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23 (1986) 485–507. CrossRef