“…Two decades ago, for the discretization in time of the nonlinear Schrödinger (NLS) equation, C. Besse [4] introduced a new linear-implicit, conservative time-stepping method (called Relaxation Scheme (RS)) as an attempt to avoid the numerical solution of the nonlinear systems of algebraic equations that the application of the implicit Crank-Nicolson method yields. The (RS) combined with a finite element or a finite difference space discretization, is computationally efficient (see, e.g., [3,8,11]) and performs as a second order method (see, e.g., [5,11]). Later, C. Besse [5], analysing the (RS), as a semidiscrete in time method to approximate the solution to the Cauchy problem for the (NLS) equation with power non-linearity, provided its convergence under small final time , without concluding a convergent rate with respect to the time-step.…”