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

Abstract

The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrödinger equation in the semiclassical regime, where the Planck constant ε is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrödinger equation with a linear potential. The local discretization error of the two time-splitting methods constructed here is O(max⁡{Δt3,Δt5/ε}), while the well-known Lie-Trotter splitting scheme and the Strang splitting scheme are O(Δt2/ε) and O(Δt3/ε), respectively, where Δt 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 Schrödinger equation with a quadratic potential.