Universality of the 11/4 order-reduction barrier

Determine whether convergence order 11/4 is a strict upper bound for every explicit exponential Runge–Kutta method of order at least three applied to problems of the form u'(t)+Au(t)=Bu(t) with non-commuting unbounded operators A and B, or whether it is only a consequence of the particular fourth- and third-order stiff order conditions analyzed in the paper and its cited predecessor.

Background

The paper analyzes four-stage, fourth-order explicit exponential Runge–Kutta methods for linear parabolic problems with non-commuting unbounded operators. Under the weak order conditions considered, the analysis and numerical experiments yield an observed convergence order of 11/4 rather than the classical fourth order.

The authors explicitly leave unresolved whether this fractional order is an inherent limitation for all explicit exponential Runge–Kutta methods of order three or higher in the non-commuting setting, or whether alternative choices of fourth-order order conditions can produce different convergence orders. Resolving this issue would establish whether the 11/4 rate is a universal structural ceiling or merely method-dependent.