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.

翻译：本文研究了两类新的四阶段四阶显式指数龙格-库塔方法，用于求解刚性或高振荡系统$y'(t)+My(t)=f(y(t))$。通过对比数值解与精确解的泰勒展开，我们推导了这些新显式指数方法的阶条件，其与经典显式龙格-库塔方法的阶条件完全一致；当$M\rightarrow \mathbf{0}$时，这些指数方法退化为经典龙格-库塔方法。进一步地，我们详细分析了这些新指数方法的线性稳定性与收敛性。最后，通过多个数值算例表明，在求解刚性系统或高振荡问题时，这些新指数方法相比标准指数积分器具有更高的精度与效率。