High order efficient splittings for the semiclassical time–dependent Schrödinger equation

Abstract

Standard numerical schemes with time–step h deteriorate (e.g. like ε−2h2) in the presence of a small semiclassical parameter ε in the time–dependent Schrödinger equation. The recently introduced semiclassical splitting was shown to be of order O(εh2). We present now an algorithm that is of order O(εh7+ε2h6+ε3h4) at the expense of roughly three times the computational effort of the semiclassical splitting and another that is of order O(εh6+ε2h4) at the same expense of the computational effort of the semiclassical splitting.