ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

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

Dahlby, Mortena1 and Owren, Brynjulfa1

a1 Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway. dahlby@math.ntnu.no; brynjulf.owren@math.ntnu.no

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.

(Received August 26 2008)

(Online publication July 8 2009)

Key Words:

  • Finite difference method;
  • stability;
  • energy conservation;
  • nonlinear Schrödinger equation;
  • linearly implicit methods.

Mathematics Subject Classification:

  • 65M10;
  • 35Q55
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