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In this paper, two new families of fourth-order explicit exponentialRunge-Kutta methods with four stages are studied for solving stiff or highlyoscillatory systems $y'(t)+My(t)=f(y(t))$. By comparing the Taylor expansionsof numerical and exact solutions, we derive the order conditions of these newexplicit exponential methods, which are exactly identical to the orderconditions of the classical explicit Runge-Kutta methods, and these exponentialmethods reduce to the classical Runge-Kutta methods once $M\rightarrow\mathbf{0}$. Furthermore, we analyze the linear stability properties and theconvergence of these new exponential methods in detail. Finally, severalnumerical examples are carried out to illustrate the accuracy and efficiency ofthese new exponential methods when applied to the stiff systems or highlyoscillatory problems than standard exponential integrators.

Two new families of fourth-order explicit exponential Runge-Kutta methods with four stages for stiff or highly oscillatory systems