DeC and ADER: Similarities, Differences and an Unified Framework

In this paper, we demonstrate that the 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. Finally, we can also investigate the stability regions for the ADER approach for different orders using several basis functions and compare them with the DeC ansatz. [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.

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