High-order commutator-free Magnus integrators and related methods for non-autonomous linear evolution equations

The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Due to the fact that previously proposed commutatorfree Magnus integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised partial differential equations of parabolic type or for master equations describing dissipative quantum systems, respectively. In order to remedy this issue, two different approaches for the design of efficient time integrators of orders four, five, and six are pursued: (i) the study of commutator-free Magnus integrators involving complex coefficients that satisfy a positivity condition, and (ii) the study of unconventional methods in the sense that an additional exponential involving a commutator of higher order with respect to the time stepsize occurs. Numerical experiments for a test equation of parabolic type and a (dissipative) Rosen–Zener model confirm that the identified novel time integrators are superior to commutator-free Magnus integrators previously proposed in the literature.