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

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