Applications of Mathematics, first online, pp. 1-20


A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang, Jun-Wei Jin, Hai-Wei Sun, Qin Sheng

Received December 4, 2019.   Published online December 16, 2020.

Abstract:  We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma$ in time, where $0<\gamma<1$ is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.
Keywords:  nonlinear time fractional Schrödinger equations; $L1$ formula; hybrid compact difference method; linearization; unconditional stability
Classification MSC:  65M06, 65M20, 65M60
DOI:  10.21136/AM.2020.0339-19

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Affiliations:   Chun-Hua Zhang, Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China, e-mail: chzringlang@163.com; Jun-Wei Jin, Key Laboratory of Grain Information Processing and Control, Ministry of Education of China, and College of Information Science and Engineering, Henan University of Technology, Zhengzhou 450001, China, e-mail: jinjunwei24@163.com; Hai-Wei Sun, Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China, e-mail: hsun@um.edu.mo; Qin Sheng (corresponding author), Department of Mathematics and Center for Astrophysics, Space Physics and Engineering Research, Baylor University, Waco, TX 76798-7328, USA, e-mail: Qin_Sheng@baylor.edu


 
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