Spectral approximation for nonlinear time fractional Schrödinger equation on graded meshes

ABSTRACT

This paper is concerned with the numerical approximation of the nonlinear time fractional Schrödinger equation subject to the initial and Neumann boundary conditions whose solution exhibits an initial weak singularity. A linearized fully discrete scheme is presented by using the finite difference method on graded meshes for temporal discretization and Gauss Lobatto Legendre Birkhoff spectral method for spatial discretization. Based on a temporal-spatial error splitting argument, the boundedness of the numerical solution in the $L^{\mathrm{\infty }}$ norm is proved rigorously. The convergence of the proposed scheme is obtained unconditionally. The theoretical results are verified through some numerical examples.

KEYWORDS:

• Time fractional Schrödinger equation
• weak singularity
• spectral method
• finite difference method
• convergence

AMS CLASSIFICATIONS:

• 35R11
• 65M12
• 65M70

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant number 11672011].