Numerical Solution of Time-Fractional Schrödinger Equation by Using FDM

Abstract

In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo derivative, and the fourth-order compact finite difference scheme is utilized for discretizing the spatial term. The unconditional stability and convergence of the scheme in the maximum norm are proved. Finally, we verified the theoretical result with a numerical test.

Keywords

• time-fractional Schrödinger equation
• L1-2-3 formula
• compact finite difference method
• stability
• Caputo derivative