A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation Original Research Article

A generalization of the Crank–Nicolson algorithm to higher orders for the time-dependent Schrödinger equation is proposed to improve the accuracy of the time approximation. The implicit difference schemes are obtained in terms of the Magnus expansion for the evolution operator and its further factorization with the help of diagonal Padé approximations. Stability of the schemes and conservation of the approximated solution norm are provided by the fact that the Magnus expansion of the evolution operator preserves its unitarity in any order with respect to a time step τ. As an example, a comparison between the numerical and analytical solutions of a model problem for the oscillator with an explicitly time-dependent frequency was performed for the schemes O(τ4) and O(τ6) to demonstrate accuracy, efficiency and adequate convergence of the method.