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

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.

An Algorithm for Computing Coefficients of Words in Expressions Involving Exponentials and Its Application to the Construction of Exponential Integrators

“…with coefficients a j,k given in Table 3. For these coefficients the positivity condition is not satisfied, in agreement with the fact that this cannot be the case for real coefficients, see [10]. For some applications, however, it is essential for stability reasons that this condition is satisfied.…”

Non-existence of generalized splitting methods with positive coefficients of order higher than four

“…A proof of Theorem 2 was proposed in [7]. In Section 2 we will give a new independent proof by showing that Theorem 3 (and thus also Theorem 2) is an easy consequence of a recent result proved by the authors in [8].…”

“…2 Note that we have changed some denotations: H(t), H 0 , H 1 , s, a j , c j correspond respectively to the denotations A(t), A 0 , A 1 , J, b j , y j of [8].…”

“…The essential step leading to the main result of [8] is comprised by the following proposition. 2 Proposition 1.…”

“…Remark 2. Strictly speaking, only a version of Proposition 1 with the weaker conclusion that at least one of the coefficients a j is non-positive has been proved in [8]. Since we may assume form the outset that a j = 0 for j = 2, .…”