Tenth-Order Accurate Numerical Method for Solving the Time-Dependent Schrödinger Equation

A tenth-order accurate method for the numerical solution of the time-dependent Schrödinger equation is presented. The method is based on the approximation of the evolution operator by a product formula. A decrease in the number of operator exponentials in the resulting formula due to the optimization of their sequence is discussed. Based on the idea proposed by Yoshida, two tenth-order accurate algorithms for approximating the evolution operator are constructed. Numerical tests demonstrate the stability and the order of accuracy of these algorithms. The method used in the paper considerably reduces the number of exponential multipliers in the scheme as compared with the well-known Lie–Trotter–Suzuki formula.