Two new families of fourth-order explicit exponential Runge--Kutta methods with four stages for first-order differential systems

In this paper, two new families of fourth-order explicit exponential Runge--Kutta (ERK) methods with four stages are studied for solving first-order differential systems $y'(t)+My(t)=f(y(t))$. By comparing the Taylor series of the exact solution, the order conditions of these ERK methods are derived, which are exactly identical to the order conditions of explicit Runge--Kutta methods, and these ERK methods reduce to classical Runge--Kutta methods once $M\rightarrow \mathbf{0}$. Moreover, we analyze the stability properties and the convergence of the new methods. Several numerical examples are implemented to illustrate the accuracy and efficiency of these ERK methods by comparison with standard exponential integrators.