  The Deferred Correction (DeC) is an iterative procedure, characterized by
increasing accuracy at each iteration, which can be used to design numerical
methods for systems of ODEs. The main advantage of such framework is the
automatic way of getting arbitrarily high order methods, which can be put in
Runge--Kutta (RK) form. The drawback is the larger computational cost with
respect to the most used RK methods. To reduce such cost, in an explicit
setting, we propose an efficient modification: we introduce interpolation
processes between the DeC iterations, decreasing the computational cost
associated to the low order ones. We provide the Butcher tableaux of the new
modified methods and we study their stability, showing that in some cases the
computational advantage does not affect the stability. The flexibility of the
novel modification allows nontrivial applications to PDEs and construction of
adaptive methods. The good performances of the introduced methods are broadly
tested on several benchmarks both in ODE and PDE contexts.
