Optimal error analysis of Crank–Nicolson schemes for a coupled nonlinear Schrödinger system in 3D

The paper is concerned with the time step condition of the commonly-used semi-implicit Crank–Nicolson finite difference schemes for a coupled nonlinear Schrödinger system in three dimensional space. We present the optimal L² error estimate without any restriction on time step, while all previous works require certain time step conditions. Our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate (inequality) of the error function. Numerical examples for both two-dimensional and three-dimensional models are investigated and numerical results illustrate our theoretical analysis.