论文标题
DEC和ADER：相似性，差异和统一框架
DeC and ADER: Similarities, Differences and a Unified Framework
论文作者
论文摘要
在本文中，我们证明了[1]中使用的明确ADER方法被视为对[2]中介绍的递延校正（DEC）方法的特殊解释。通过使用这一事实，我们能够将ADER嵌入时间集成方案的理论背景中，并证明准确性顺序与达到所需顺序所需的迭代次数之间的关系。接下来，我们将调查扩展到僵硬的ODE，隐含地处理这些源术语。可以找到解释和实施的一些差异。使用DEC的产生通常是一个简单得多的实现，而ADER至少在我们的数值模拟中受益于更高的精度。然后，我们还专注于PDE案例，并使用封闭形式的DEC和ADER提出常见的时空离散。最后，在数值部分中，我们研究了ADER方法的A稳定性 - 这是第一次与我们的知识进行 - 使用多个基础函数进行不同的顺序，并将其与DEC ANSATZ进行比较。然后，我们比较ADER和DEC的僵硬和非建筑odes的性能，并验证我们的分析重点是两个基本的双曲线问题。 [1] O. Zanotti，F。Fambri，M。Dumbser和A. Hidalgo。时空自适应ADER不连续的Galerkin有限元方案具有后部子细胞有限体积限制。计算机与流体，118：204-224，2015。 [2] A. Dutt，L。Greengard和V. Rokhlin。普通微分方程的光谱递延校正方法。位数学数学，40（2）：241-266，2000。
In this paper, we demonstrate that the explicit ADER approach as it is used inter alia in [1] can be seen as a special interpretation of the deferred correction (DeC) method as introduced in [2]. 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. [1] O. Zanotti, F. Fambri, M. Dumbser, and A. Hidalgo. Space-time adaptive ader discontinuous galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Computers & Fluids, 118:204-224, 2015. [2] A. Dutt, L. Greengard, and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics, 40(2):241-266, 2000.
