In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge--Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general format of so-called modified two-derivative diagonally implicit Runge--Kutta methods (TDDIRK) is proposed. Their order conditions up to order six are derived by introducing a set of bi-coloured rooted trees and deriving new elementary weights. Next, we build exponential fitting conditions in order for these modified TDDIRK methods to treat oscillatory solutions, leading to EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can be considered as superconvergent methods. The stability and phase-lag analysis of the new methods are also investigated, leading to optimized fourth-order schemes, which turn out to be much more accurate and efficient than their non-optimized versions. Finally, we carry out numerical experiments on some oscillatory test problems. Our numerical results clearly demonstrate the accuracy and efficiency of the newly derived methods when compared with existing implicit Runge--Kutta methods and two-derivative Runge--Kutta methods of the same order in the literature.

翻译：在这项工作中,我们建造并开发出一个新的类别,用双色根植树和产生新的基本重量,来制造和产生一个具有双色底植树和六级的两代底植树(EFTDDIRK)的指数性双向隐含龙格-库塔(EFTDDIRK)方法(EFTDDIRK)方法,用于用螺旋形解析方程式解决不同方程式的数值解决方案。首先,提出了所谓的修改的两代半半隐含龙格-库塔方法(TDDIRK)的一般格式。这些方法的顺序条件可以被认为是超级趋同的方法。对新方法的稳定性和阶段级分析也进行了调查,从而优化了第四级方案,这些方法比这些经过修改的TDDIRITK方法更准确和高效,从而导致采用EFTDDIRRK方法。最后,我们用目前两种不源取序的流程方法来明确展示了我们当前排序方法的运行效率。