Numerical study of fourth-order linearized compact schemes for generalized NLS equations

The fourth-order compact approximation for the spatial second-derivative and several linearized approaches, including the time-lagging method of Zhang et al. (1995), the local-extrapolation technique of Chang et al. (1999) and the recent scheme of Dahlby et al. (2009), are considered in constructing fourth-order linearized compact difference (FLCD) schemes for generalized NLS equations. By applying a new time-lagging linearized approach, we propose a symmetric fourth-order linearized compact difference (SFLCD) scheme, which is shown to be more robust in long-time simulations of plane wave, breather, periodic traveling-wave and solitary wave solutions. Numerical experiments suggest that the SFLCD scheme is a little more accurate than some other FLCD schemes and the split-step compact difference scheme of Dehghan and Taleei (2010). Compared with the time-splitting pseudospectral method of Bao et al. (2003), our SFLCD method is more suitable for oscillating solutions or the problems with a rapidly varying potential.