A new second-order accurate time discretization method for the nonlinear cubic Schrodinger equation

作者：Samrout Y M^*

来源：International Journal of Computer Mathematics, 2010, 87(13): 2967-2997.

DOI：10.1080/00207160902874646

• nonlinear cubic Schrodinger equation
• the method of lines
• the Lanczos' Tau method
• the Crank-Nicolson method
• the Newton method

A new second-order accurate semi-analytical time discretization method is introduced for the numerical solution of the one-dimensional nonlinear cubic Schrodinger equation. This method is based on the combination of the method of lines, Crank-Nicolson method, Newton method and Lanczos' Tau method. It is a self-starting averaged two-time-level scheme that has proved to be stable, accurate and energy conservative for long time integration periods. At each time level, approximate solutions are sought on a segmented spatial interval as finite expansions in terms of a given orthogonal polynomial basis mapped appropriately onto each spatial subsegment. We have carried out numerical simulation concerning several cases for the propagation, collision and the bound states of solitons. Accurate results have been obtained using Chebyshev and Legendre polynomials. These results are well comparable with other published results obtained by the use of various standard numerical methods.

• 出版日期2010