Split-step spectral Galerkin method for the two-dimensional nonlinear space-fractional Schrödinger equation

Abstract

In this paper, we propose a highly accurate and conservative split-step spectral Galerkin (SSSG) scheme, which combines the standard second-order Strang split-step method for handling the nonlinear and the potential terms with the Legendre spectral Galerkin method for approximating the Riesz space-fractional derivatives, for the two-dimensional nonlinear space-fractional Schrödinger equation. The mass conservation property of the numerical solution is proved and the optimal error estimate with respect to spatial discretization is established by introducing an orthogonal projection operator. In addition, a matrix diagonalization technique is introduced to resolve the multi-dimensional difficulty and reduce the computational complexity in the implementation. Numerical experiments are presented to illustrate the accuracy and robustness of the SSSG method.