Abstract
In this paper, we develop an implicit difference method for solving the nonlinear time-space fractional Schrödinger equation. The scheme is constructed by using the L2-1σ formula to approximate the Caputo fractional derivative, while the weighted and shifted Grünwald 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.
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