DeC and ADER: Similarities, Differences and a Unified Framework

Abstract

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.