This paper deals with the numerical solution of both linear and non-linear Schrödinger problems, which mathematically model many physical processes in a wide range of applications of interest. In particular, a comparison of different solvers and different approaches for these problems is developed throughout this work. Two finite difference schemes are analyzed: the classical Crank-Nicolson approach, and a high-order compact scheme. Solvers based on geometric multigrid, Fast Fourier Transform and Alternating Direction Implicit methods are compared. Finally, the efficiency of the considered solvers is tested for a linear Schrödinger problem, proving that the computational experiments are in good agreement with the theoretical predictions. In order to test the robustness of the MG solver two additional Schrödinger problems with a nonconstant potential and nonlinear right-hand side are solved by the MG solver, since the efficiency of this solver depends on such data.