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The Deferred Correction (DeC) is an iterative procedure, characterized byincreasing accuracy at each iteration, which can be used to design numericalmethods for systems of ODEs. The main advantage of such framework is theautomatic way of getting arbitrarily high order methods, which can be put inRunge--Kutta (RK) form. The drawback is the larger computational cost withrespect to the most used RK methods. To reduce such cost, in an explicitsetting, we propose an efficient modification: we introduce interpolationprocesses between the DeC iterations, decreasing the computational costassociated to the low order ones. We provide the Butcher tableaux of the newmodified methods and we study their stability, showing that in some cases thecomputational advantage does not affect the stability. The flexibility of thenovel modification allows nontrivial applications to PDEs and construction ofadaptive methods. The good performances of the introduced methods are broadlytested on several benchmarks both in ODE and PDE contexts.

A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity