In this paper we study conditional stability properties of exponential Runge–Kutta methods when they
are applied to semilinear systems of ordinary differential equations characterized by a stiff linear part and
a nonstiff nonlinear part. In particular,we obtain sufficient conditions under which an explicit method
satisfies such conditional properties. We also study the unconditional stability properties introduced in
our previous paper (Maset & Zennaro,2008,Unconditional stability of explicit exponential Runge-Kutta
methods for semi-linear ordinary differential equations. Math. Comput.,78,957–967). In particular,we
obtain a necessary condition for such unconditional properties. By using the sufficient conditions for the
conditional properties and the necessary condition for the unconditional properties,we analyse and classify
the most popular explicit methods. The research that led to the present paper was partially supported by a grant of the group GNCS of INdAM.
