In this paper, we demonstrate that the explicit ADER approach as it is used inter alia in Zanotti et al. (Comput Fluids 118:204–224, 2015) can be seen as a special interpretation of the deferred correction (DeC) method as introduced in Dutt et al. (BIT Numer Math 40(2):241–266, 2000). By using this fact, we are able to embed ADER in a theoretical background of time integration schemes and prove the relation between the accuracy order and the number of iterations which are needed to reach the desired order. Next, we extend our investigation to stiff ODEs, treating these source terms implicitly. Some differences in the interpretation and implementation can be found. Using DeC yields typically a much simpler implementation, while ADER benefits from a higher accuracy, at least for our numerical simulations. Then, we also focus on the PDE case and present common space-time discretizations using DeC and ADER in closed forms. Finally, in the numerical section we investigate A-stability for the ADER approach—this is done for the first time up to our knowledge—for different order using several basis functions and compare them with the DeC ansatz. Then, we compare the performance of ADER and DeC for stiff and non-stiff ODEs and verify our analysis focusing on two basic hyperbolic problems.

在本文中，我们演示了显式ADER方法以及其他方法在Zanotti等人中。（Comput Fluids 118：204–224，2015年）可以看作是Dutt等人引入的对延迟校正（DeC）方法的特殊解释。（BIT纽默数学40（2）：241-266，2000）。利用这一事实，我们能够将ADER嵌入时间积分方案的理论背景中，并证明精度阶数与达到所需阶数所需的迭代次数之间的关系。接下来，我们将研究扩展到刚性ODE，并隐式处理这些源术语。可以在解释和实现上找到一些差异。使用DeC通常会产生更简单的实现，而ADER至少从我们的数值模拟中受益于更高的精度。然后，我们还关注PDE情况，并以封闭形式使用DeC和ADER给出常见的时空离散化。最后，在数字部分中，我们研究了ADER方法的A稳定性（这是我们所知第一次），使用几个基本函数以不同的阶数进行了比较，并将它们与DeC ansatz进行了比较。然后，我们比较了ADER和DeC对刚性和非刚性ODE的性能，并验证了我们针对两个基本双曲问题的分析。