DeC and ADER: Similarities, Differences and a Unified Framework

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