Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation

Abstract In this article, we consider the two‐dimensional nonlinear time–space fractional Schrödinger equation with space described by the fractional Laplacian. A second‐order fractional backward difference formula in the temporal direction while Fourier spectral method in the spatial direction is proposed to solve the model numerically. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. By using the generalized discrete Gronwall inequality developed by Dixon and McKee and the temporal–spatial error splitting argument, the convergence of the fast time‐stepping numerical method is also proved in a simple manner without imposing the Courant‐Friedrichs‐Lewy (CFL) condition. Finally, some numerical results are provided to support the theoretical analysis.