ORIGINAL PAPER Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schro ¨ dinger equations Mustafa Almushaira 1,2,3 • Fei Liu 1,2 Received: 15 May 2020 / Accepted: 6 October 2020 Ó Springer Nature Switzerland AG 2020 Abstract In this work, an efficient fourth-order time-stepping compact finite difference scheme is devised for the numerical solution of multi-dimensional space-fractional coupled nonlinear Schro ¨dinger equations. Some existing numerical schemes for these equations lead to full and dense matrices due to the non-locality of the fractional operator. To overcome this challenge, the spatial discretization in our method is carried out by using the compact finite difference scheme and matrix transfer technique in which FFT-based computations can be utilized. This avoids storing the large matrix from discretizing the fractional operator and also significantly reduces the computational costs. The amplification symbol of this scheme is investigated by plotting its stability regions, which indicates the stability of the scheme. Numerical experiments show that this scheme preserves the conservation laws of mass and energy, and achieves the fourth-order accuracy in both space and time. Keywords Space-fractional nonlinear Schro ¨dinger equations  Time-stepping methods  Matrix transfer technique  Discrete sine transform Mathematics Subject Classification 35Q41  35R11  65M06  65M12 1 Introduction In this paper, we consider the following multi-dimensional space-fractional coupled nonlinear Schro ¨dinger (SFCNLS) equations involving the fractional Laplacian ðDÞ a=2 ð1\a  2Þ This article is part of the section ‘‘Computational Approaches’’ edited by Siddhartha Mishra. & Fei Liu liufei@hust.edu.cn 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China 3 Department of Mathematics, Faculty of Science, Sana’a University, Sanaa, Yemen SN Partial Differential Equations and Applications SN Partial Differ. Equ. Appl. (2020)1:45 https://doi.org/10.1007/s42985-020-00048-6