Error analysis of a fully discrete scheme for time fractional Schrödinger equation with initial singularity

We consider the numerical approximation for a time fractional Schrödinger equation whose solution exhibits an initial weak singularity. A fully discrete scheme is constructed using scheme on graded mesh for the discretiaztion of temporal Caputo derivative and spectral method for spatial discretization. It is shown that with appropriate choice of the grading parameter, the scheme can attain order convergence in temporal direction, where α is the order of time Caputo fractional derivative, and spectral accuracy in spatial direction if the solution is sufficiently smooth in its spatial part. Numerical results confirm the sharpness of the error analysis.

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