The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient $p$-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization.

翻译：（现代）任意导数（ADER）方法是一种基于迭代求解弱形式隐式离散化的微分问题数值求解技术。本文以常微分方程（ODE）为例，研究改进该方法的若干策略。我们首先探讨该方法精度阶次与弱形式多项式离散化之间的联系，证明精确参数选择可使其收敛阶数高于现有文献结果。随后，我们将ADER方法纳入延迟校正（DeC）框架，据此确定最优迭代次数（等于方法的理论精度阶数），并引入高效的p自适应修正方案。该方案通过匹配每次迭代中实际达到的精度阶次与多项式重构次数实现自适应调整。我们提供理论分析与数值实验结果，包括新型修正方法的稳定性分析、计算效率研究、自适应应用实例，以及基于谱差分（SD）空间离散化的双曲型偏微分方程应用实例。