A second-order implicit difference scheme for the nonlinear time-space fractional Schroedinger equation

摘要

In this paper, we develop an implicit difference method for solving the nonlinear time-space fractional Schrodinger equation. The scheme is constructed by using the L2-1_σ formula to approximate the Caputo fractional derivative, while the weighted and shifted Gruenwald formula is adopted for the spatial discretization. The stability and unique solvability of the difference scheme are analyzed in detail. Moreover, we prove that the numerical solution is convergent with second-order accuracy in both temporal and spatial directions. Finally, a linearized iterative algorithm is provided and some numerical tests are presented to validate our theoretical results.