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微分问题弱形式隐式离散化的流行数值技术。本文以常微分方程(ODE)为背景，研究了改进该方法的若干策略。我们首先关注弱形式多项式离散化与精度的关系，证明特定选择可较现有文献实现更高阶收敛。随后，我们将ADER方法纳入延迟修正(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——通过匹配每次迭代的实际精度阶数与多项式重构次数实现。我们提供了分析与数值结果，包括新修正方法的稳定性分析、计算效率研究、自适应应用实例，以及采用谱差分(SD)空间离散化的双曲型偏微分方程(PDE)应用。