Necessity of an additional stage to prevent order reduction

Determine whether adding an extra stage to an explicit exponential Runge–Kutta method is necessary to avoid order reduction for problems with non-commuting unbounded operators A and B, even when B satisfies stronger regularity assumptions.

Background

The analyzed four-stage, fourth-order schemes exhibit order reduction to approximately 11/4 when the stiff and perturbation operators do not commute. The conclusion raises the possibility that increasing the number of stages may be required to recover full fourth-order convergence.

The unresolved issue is whether an additional stage is genuinely necessary, including in settings where the perturbation operator B has stronger regularity than assumed in the paper. This question is directly connected to the design of exponential Runge–Kutta methods that retain their nominal order in non-commuting problems.