On high-order compact schemes for the multidimensional time-fractional Schrödinger equation

Abstract

Abstract In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O ( τ 2 − α + h 4 ) $O(\tau ^{2-\alpha }+h^{4})$ and O ( τ 3 − α + h 4 ) $O(\tau ^{3-\alpha }+h^{4})$ , respectively, with the temporal step size τ and the spatial step size h, and α ∈ ( 0 , 1 ) $\alpha \in (0,1)$ . For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

Keywords

• Time-fractional Schrödinger equation
• L1-2 and L1 formulas
• Compact finite difference method
• ADI
• Stability