Plane wave stability of some conservative schemes for the cubic Schrödinger equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, p. 677-687

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.

DOI : https://doi.org/10.1051/m2an/2009022
Classification:  65M10,  35Q55
Keywords: finite difference method, stability, energy conservation, nonlinear Schrödinger equation, linearly implicit methods
@article{M2AN_2009__43_4_677_0,
     author = {Dahlby, Morten and Owren, Brynjulf},
     title = {Plane wave stability of some conservative schemes for the cubic Schr\"odinger equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {4},
     year = {2009},
     pages = {677-687},
     doi = {10.1051/m2an/2009022},
     zbl = {1167.65449},
     mrnumber = {2542871},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_4_677_0}
}
Dahlby, Morten; Owren, Brynjulf. Plane wave stability of some conservative schemes for the cubic Schrödinger equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, pp. 677-687. doi : 10.1051/m2an/2009022. http://www.numdam.org/item/M2AN_2009__43_4_677_0/

[1] M.J. Ablowitz and J.F. Ladik, A nonlinear difference scheme and inverse scattering. Studies Appl. Math. 55 (1976) 213-229. | MR 471341 | Zbl 0338.35002

[2] H. Berland, B. Owren and B. Skaflestad, Solving the nonlinear Schrödinger equation using exponential integrators. Model. Ident. Control 27 (2006) 201-218.

[3] C. Besse, A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42 (2004) 934-952 (electronic). | MR 2112787 | Zbl 1077.65103

[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. | MR 2413146 | Zbl 1147.65102

[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. | MR 1752264 | Zbl 0954.65087

[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. | MR 1343329 | Zbl 0832.65136

[7] 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). | MR 2221614 | Zbl 1094.65125

[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. | MR 1857623 | Zbl 0989.65102

[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. | MR 1848726 | Zbl 0993.65098

[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. | MR 762363 | Zbl 0541.65082

[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. | MR 842641 | Zbl 0597.76012