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We propose a new paradigm for designing efficient p-adaptive arbitrary highorder methods. We consider arbitrary high order iterative schemes that gain oneorder of accuracy at each iteration and we modify them in order to match theaccuracy achieved in a specific iteration with the discretization accuracy ofthe same iteration. Apart from the computational advantage, the new modifiedmethods allow to naturally perform p-adaptivity, stopping the iterations whenappropriate conditions are met. Moreover, the modification is very easy to beincluded in an existing implementation of an arbitrary high order iterativescheme and it does not ruin the possibility of parallelization, if this wasachievable by the original method. An application to the Arbitrary DERivative(ADER) method for hyperbolic Partial Differential Equations (PDEs) is presentedhere. We explain how such framework can be interpreted as an arbitrary highorder iterative scheme, by recasting it as a Deferred Correction (DeC) method,and how to easily modify it to obtain a more efficient formulation, in which alocal a posteriori limiter can be naturally integrated leading to p-adaptivityand structure preserving properties. Finally, the novel approach is extensivelytested against classical benchmarks for compressible gas dynamics to show therobustness and the computational efficiency.

Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order