In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a first-order symplectic method. In order to solve highly oscillatory problems, the highly accurate implicit ERK integrators (up to order four) are formulated by comparing the Taylor expansions of numerical and exact solutions, it is shown that the order conditions of two new kinds of exponential methods are identical to the order conditions of classical Runge-Kutta (RK) methods. Moreover, we investigate the linear stability properties of these exponential methods. Finally, numerical results not only present the long time energy preservation of the first-order symplectic method, but also illustrate the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.

翻译：本文研究了求解高振荡系统的两类新型隐式指数龙格-库塔(ERK)方法。首先，我们分析了两类指数积分器的辛条件，并提出了一类一阶辛方法。为解决高振荡问题，通过比较数值解与精确解的泰勒展开，构造了高精度隐式ERK积分器（最高达四阶），结果表明两类新型指数方法的阶条件与经典龙格-库塔(RK)方法的阶条件一致。此外，我们探讨了这些指数方法的线性稳定性特性。最后，数值算例不仅展示了一阶辛方法在长时间内保持能量守恒的特性，同时通过与标准ERK方法的对比，验证了所构造方法的精度与效率。