# Plane wave stability of some conservative schemes for the cubic Schrödinger equation

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 4, page 677-687
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topDahlby, Morten, and Owren, Brynjulf. "Plane wave stability of some conservative schemes for the cubic Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 677-687. <http://eudml.org/doc/250590>.

@article{Dahlby2009,

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.
},

author = {Dahlby, Morten, Owren, Brynjulf},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite difference method; stability; energy conservation; nonlinear Schrödinger equation; linearly implicit methods.; finite difference method; linearly implicit methods},

language = {eng},

month = {7},

number = {4},

pages = {677-687},

publisher = {EDP Sciences},

title = {Plane wave stability of some conservative schemes for the cubic Schrödinger equation},

url = {http://eudml.org/doc/250590},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Dahlby, Morten

AU - Owren, Brynjulf

TI - Plane wave stability of some conservative schemes for the cubic Schrödinger equation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/7//

PB - EDP Sciences

VL - 43

IS - 4

SP - 677

EP - 687

AB -
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.

LA - eng

KW - Finite difference method; stability; energy conservation; nonlinear Schrödinger equation; linearly implicit methods.; finite difference method; linearly implicit methods

UR - http://eudml.org/doc/250590

ER -

## References

top- M.J. Ablowitz and J.F. Ladik, A nonlinear difference scheme and inverse scattering. Studies Appl. Math.55 (1976) 213–229. Zbl0338.35002
- H. Berland, B. Owren and B. Skaflestad, Solving the nonlinear Schrödinger equation using exponential integrators. Model. Ident. Control27 (2006) 201–218.
- C. Besse, A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal.42 (2004) 934–952 (electronic). Zbl1077.65103
- E. Celledoni, D. Cohen and B. Owren, Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math.8 (2008) 303–317. Zbl1147.65102
- A. Durán and J.M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation. IMA J. Numer. Anal.20 (2000) 235–261. Zbl0954.65087
- Z. Fei, V.M. Pérez-García and L. Vázquez, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput.71 (1995) 165–177. Zbl0832.65136
- E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration, Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics31. Second Edition, Springer-Verlag, Berlin (2006). Zbl1094.65125
- A.L. Islas, D.A. Karpeev and C.M. Schober, Geometric integrators for the nonlinear Schrödinger equation. J. Comput. Phys.173 (2001) 116–148. Zbl0989.65102
- T. Matsuo and D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys.171 (2001) 425–447. Zbl0993.65098
- T.R. Taha and J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. J. Comput. Phys.55 (1984) 203–230. Zbl0541.65082
- J.A.C. Weideman and B.M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal.23 (1986) 485–507. Zbl0597.76012