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 expansion 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 properties for these exponential methods. Finally, numerical results not only display the long time energy preservation of the symplectic method, but also present 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，我们分析了两类指数积分器的辛条件，并得到了辛方法。为了有效求解高振荡问题，我们尝试设计高精度隐式ERK积分器。通过将数值解的泰勒级数展开与精确解进行比较，可以验证两类新型指数方法的阶条件与经典龙格-库塔（RK）方法相同，这意味着利用RK方法的系数可直接构建一些高精度数值方法。此外，我们还研究了这些指数方法的线性稳定性性质。最后，数值结果不仅展示了辛方法的长时间能量保持性，还与标准ERK方法相比，呈现了所构建方法的精度和效率。