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 stiff or highly oscillatory systems $y'(t)+My(t)=f(y(t))$.We analyze modified and simplified versions of fourth-order explicit exponential Runge-Kutta methods, respectively, which are different from standard exponential Runge-Kutta methods. Using the Taylor series of numerical and exact solutions, we obtain the order conditions of these new explicit exponential methods, which reduce to those of the standard Runge-Kutta methods when $M \rightarrow 0$. We show the convergence of these new exponential methods in detail. Numerical experiments are carried out, and the numerical results demonstrate the accuracy and efficiency of these new exponential methods when applied to the stiff systems or highly oscillatory problems.