Efficient exponential splitting spectral methods for linear Schrodinger equation in the semiclassical regime

The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrodinger equation in the semiclassical regime, where the Planck constant epsilon is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrodinger equation with a linear potential. The local discretization error of the two time-splitting methods constructed here is O(max{Delta t(3), Delta t(5)/epsilon}), while the well-known Lie-Trotter splitting scheme and the Strang splitting scheme are O(Delta t(2)/epsilon) and O(Delta t(3)/epsilon), respectively, where Delta t is the time step-size. The global error estimates of new exponential splitting schemes with spectral discretization suggests that larger time step-size is admissible for obtaining high accuracy approximation of the solutions. Numerical studies verify our theoretical results and reveal that the new methods are especially efficient for linear semiclassical Schrodinger equation with a quadratic potential. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.