Abstract
In this paper, firstly, we solve the linear 3D Schrödinger equation using Douglas–Gunn alternating direction implicit (ADI) scheme and high-order compact (HOC) ADI scheme, which have the order O(tau^{2}+h^{2}) and O(tau^{2}+h^{4}), respectively. Secondly, a fourth-order compact ADI scheme, based on the Douglas–Gunn ADI scheme combined with second-order Strang splitting technique, is proposed for solving 3D nonlinear Schrödinger equation. Stability analysis has demonstrated that these schemes are unconditionally stable. Finally, numerical results show that these schemes preserve the conservation laws and provide accurate and stable solutions for the 3D linear and nonlinear Schrödinger equations.
Highlights
The nonlinear Schrödinger (NLS) equation has been used extensively in underwater acoustics, quantum mechanics, plasma physics, nonlinear optics, electromagnetic wave propagation, etc. [1,2,3,4]
The rest of our paper is organized as follows: In Sect. 2, we present a standard Douglas– Gunn alternating direction implicit (ADI) scheme and a new high-order compact (HOC)-ADI scheme for the 3D LS equation, and the stability of the standard Douglas–Gunn ADI scheme and the new HOC-ADI scheme is investigated
We investigate the stability of D-G ADI scheme using the Fourier analysis method
Summary
The nonlinear Schrödinger (NLS) equation has been used extensively in underwater acoustics, quantum mechanics, plasma physics, nonlinear optics, electromagnetic wave propagation, etc. [1,2,3,4]. Gao and Xie [24] proposed a fourth-order ADI compact finite difference scheme for two-dimensional Schrödinger equation. Liao et al [25] established a compact ADI scheme for solving linear Schrödinger equations These three articles are second-order in time and fourth-order in space with less computational cost. Introducing the intermediate variables, we obtain the new HOC-ADI scheme aiδx u∗jkl = τ ai LyLzδx2 + LxLzδy2 + LxLyδz unjkl,. For this method, intermediate values of u∗∗ and u∗ at the boundary are obtained by (2.10c) and (2.10b). Proof Eliminating the intermediate variables in the new HOC-ADI scheme (2.10a)– (2.10d), we have unjk+l 1 + unjkl. Theorem 3 The D-G ADI scheme (2.6a)–(2.6d) and the new HOC-ADI scheme (2.10a)– (2.10d) are unconditionally stable
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