In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. Firstly, we analyze the symplectic conditions for two kinds of exponential integrators and obtain the symplectic method. In order to effectively solve highly oscillatory problems, we try to design the highly accurate implicit ERK integrators. By comparing the Taylor series of numerical solution with exact solution, it can be verified that the order conditions of two new kinds of exponential methods are identical to classical Runge-Kutta (RK) methods, which implies that using the coefficients of RK methods, some highly accurate numerical methods are directly formulated. Furthermore, we also investigate the linear stability regions for these exponential methods. Finally, numerical results not only display the long time energy preservation of the symplectic method, but also illustrate the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.

翻译：本文研究了两类新型隐式指数龙格-库塔(ERK)方法，用于求解高振荡系统。首先，我们分析了两种指数积分器的辛条件，并得到了辛方法。为有效求解高振荡问题，我们尝试设计高精度隐式ERK积分器。通过比较数值解与精确解的泰勒级数，可验证两类新型指数方法的阶条件与经典龙格-库塔(RK)方法一致，这意味着利用RK方法的系数可直接构造出若干高精度数值方法。此外，我们还研究了这些指数方法的线性稳定区域。最后，数值结果不仅展示了辛方法长期能量保持的特性，而且通过与标准ERK方法的对比，说明了所构造方法的精度和效率。