A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

In this paper, based on Galerkin–Legendre spectral method for space discretization and a linearized Crank–Nicolson difference scheme in time, a fully discrete spectral scheme is developed for solving the strongly coupled nonlinear fractional Schrödinger equations. We first prove that the proposed scheme satisfies the conservation laws of mass and energy in the discrete sense. Then a prior bound of the numerical solutions in $L^{\infty }$ L ∞ -norm is obtained, and the spectral scheme is shown to be unconditionally convergent in $L^{2}$ L 2 -norm, with second-order accuracy in time and spectral accuracy in space. Finally, some numerical results are provided to validate our theoretical analysis.

Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations

To improve the computing efficiency, a fourth-order difference scheme is proposed and a fast algorithm is designed to simulate the nonlinear fractional Schrödinger (FNLS) equation oriented from the fractional quantum mechanics. The numerical analysis and experiments conducted in this article show that the proposed difference scheme has the optimal second-order and fourth-order convergence rates in time and space respectively, reduces its computation cost to $\mathcal{O}(M\log M)$O(MlogM), and recognizes accurately its physical feature of FNLS such as the mass balance.