High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equation

Abstract

The class of commutator-free quasi-Magnus (CFQM) exponential integrators provides a favourable alternative to standard Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. The schemes are given by compositions 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 CFQM exponential integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised parabolic equations or for master equations describing dissipative quantum systems. 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 CFQM exponential 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 confirm that the identified novel time integrators are superior to other integrators of the same family previously proposed in the literature.