On high-order compact schemes for the multidimensional time-fractional Schr\xf6dinger equation

Rena Eskar,Ehmet Kasim,Xinlong Feng

doi:10.1186/s13662-020-02946-w

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(tau ^{2-alpha }+h^{4}) and O(tau ^{3-alpha }+h^{4}), respectively, with the temporal step size τ and the spatial step size h, and 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.

Highlights

1 Introduction Nowadays fractional differential equations have been widely studied in many fields, owing to their diverse applications in physics, biology, chemistry, mechanics, and finance theory [1,2,3,4,5,6,7,8,9]
To improve the numerical accuracy, some fourth-order compact finite difference methods have been proposed for oneand two-dimensional time-fractional partial differential equations
We investigate the stability of the high-order compact alternating direction implicit (ADI) schemes (CS2DI and CS2DII) using the Fourier analysis method

Summary

Introduction

Nowadays fractional differential equations have been widely studied in many fields, owing to their diverse applications in physics, biology, chemistry, mechanics, and finance theory [1,2,3,4,5,6,7,8,9]. These applications have contributed to the emergence of various fractional differential equations in the mathematical and physical world. For two-dimensional time-fractional partial differential equations, several numerical methods have been proposed [17, 18]. To improve the numerical accuracy, some fourth-order compact finite difference methods have been proposed for oneand two-dimensional time-fractional partial differential equations.

Objectives

Conclusion