Non-satisfiability of a positivity condition for commutator-free exponential integrators of order higher than four

TL;DR

This paper proves that for commutator-free exponential integrators applied to parabolic problems, the positivity condition restricts their maximal order to four, confirming previous conjectures and impacting high-order method design.

Contribution

It establishes that positivity constraints limit the order of commutator-free exponential integrators to four for real coefficients, confirming earlier conjectures.

Findings

Positivity condition restricts integrator order to four

Maximal convergence order of four proven for real coefficients

Supports previous conjectures about integrator limitations

Abstract

We consider commutator-free exponential integrators as put forward in [Alverman, A., Fehske, H.: High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys. 230, 5930-5956 (2011)]. For parabolic problems, it is important for the well-definedness that such an integrator satisfies a positivity condition such that essentially it only proceeds forward in time. We prove that this requirement implies maximal convergence order of four for real coefficients, which has been conjectured earlier by other authors.