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.