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

Morten Dahlby; Brynjulf Owren

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

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

Abstract

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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.

How to cite

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Dahlby, 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

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  1. M.J. Ablowitz and J.F. Ladik, A nonlinear difference scheme and inverse scattering. Studies Appl. Math.55 (1976) 213–229.  Zbl0338.35002
  2. H. Berland, B. Owren and B. Skaflestad, Solving the nonlinear Schrödinger equation using exponential integrators. Model. Ident. Control27 (2006) 201–218.  
  3. C. Besse, A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal.42 (2004) 934–952 (electronic).  Zbl1077.65103
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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

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