论文标题
指数拟合的两衍生DIRK方法用于振荡微分方程
Exponentially fitted two-derivative DIRK methods for oscillatory differential equations
论文作者
论文摘要
在这项工作中，我们构建并得出了一类新的指数拟合的两源性对角线隐式runge-kutta（eftddirk）方法，用于具有振荡解决方案的微分方程的数值解。首先，提出了所谓的修改后的两衍生对角线隐式runge- kutta方法（TDDIRK）的一般格式。他们的订单条件直至六，通过引入一组双色生根树并得出新的基本权重来得出。接下来，我们构建指数拟合条件，以便这些修改后的TDDIRK方法处理振荡溶液，从而导致EFTDDIRK方法。特别是，得出了一个2阶段的四阶家族，五阶和三阶段的六阶EFTDDIRK方案。这些可以被视为超对面的方法。还研究了对新方法的稳定性和相位lag分析，从而导致了优化的四阶方案，事实证明，这些方案比其非优化版本更准确和高效。最后，我们就某些振荡性测试问题进行数值实验。我们的数值结果清楚地证明了与现有的隐式runge（kutta方法和两个衍生runge）相比，新得出的方法的准确性和效率 - 文献中相同顺序的kutta方法。
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.
