A new high-order compact ADI finite difference scheme for solving 3D nonlinear Schrödinger equation

Abstract

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(τ2+h2) $O(\tau^{2}+h^{2})$ and O(τ2+h4) $O(\tau^{2}+h^{4})$, 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.

Keywords

• 3D Schrödinger equation
• Compact ADI finite difference method
• Conservation law
• Stability
• High-order scheme