Abstract

We consider the numerical integration of semi-discretised Schrödinger equations requiring the computation of the exponential of a (skew-)Hermitian matrix acting on a vector. This is usually achieved by polynomial methods such as Taylor, Krylov or Chebyshev, which are conditionally stable. However, the skew-Hermitian matrix is usually separable into solvable parts, and tailored splitting methods can be used which preserve unitarity and they are unconditionally stable. In addition, their accuracy does not seem to deteriorate when considering a finer mesh, unlike polynomial methods. However, resonances may appear. We analyse where resonances come from and how to reduce their undesirable effects. As an alternative, we also analyse Cayley-splitting methods: they are unitary (unconditionally stable) methods, they can avoid the resonances and, in many cases, they are considerably cheaper to compute than the exponential splitting methods. Some numerical examples will illustrate the potential interest of the splitting methods as well as the new family of Cayley-splitting methods.