Symmetric-conjugate splitting methods for linear unitary problems

Abstract

We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.

Symmetric-conjugate splitting methods for evolution equations of parabolic type

“…Our investigations are inspired by a series of works on real and complex splitting methods as well as modified splitting methods. As an excerpt, we mention [1,3,4,5,6,8,9,18,19,20,21,22,23,25,27,31,32,34,36,37,38,43,44,47,48,51] and our former contributions [2,7,10,11,12,14,15,16,17,49,50], where further references are given.…”

“…Optimised symmetric schemes of order four and a symmetric scheme of order six are found in [12]. 1 Amongst the symmetric-conjugate splitting methods, we highlight sixth-order schemes, recently proposed in [7] for linear unitary problems. They set aside the second-and fourth-order barriers for standard and modified splitting methods, see [1,10,47,48,25] and references given therein.…”

“…, s}. Consequently, they are suitable for the time integration of different classes of evolution equations including parabolic and Schrödinger equations, see (6) as well as (7) below and Table 1.…”

“…Evolution equations of Schrödinger type. It is instructive to observe that linear Schrödinger equations (7) are included in (8a), see also (8c). For this reason, we may expect that the convergence analysis provided in [49,50] for Fourier spectral space discretisations combined with high-order operator splitting methods based on real coefficients can be transferred to evolution equations of parabolic type and complex symmetric-conjugate splitting methods satisfying a positivity condition.…”