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In this work, we consider the general family of the so called ADER PNPMschemes for the numerical solution of hyperbolic partial differential equationswith \textit{arbitrary} high order of accuracy in space and time.  The family of one-step PNPM schemes was introduced in [Dumbser et al., JCP,2008] and represents a unified framework for classical high order Finite Volume(FV) schemes (N=0), the usual Discontinuous Galerkin (DG) methods (N=M), aswell as a new class of intermediate hybrid schemes for which a reconstructionoperator of degree M is applied over piecewise polynomial data of degree N withM>N. In all cases with M >= N > 0 the PNPM schemes are linear in the sense ofGodunov, thus when considering phenomena characterized by discontinuities,spurious oscillations may appear and even destroy the simulation. Therefore, inthis paper we present a new simple, robust and accurate a posteriori subcellfinite volume limiting strategy that is valid for the entire class of PNPMschemes. The subcell FV limiter is activated only where it is needed, i.e. inthe neighborhood of shocks or other discontinuities, and is able to maintainthe resolution of the underlying high order PNPM schemes, due to the use of arather fine subgrid of 2N+1 subcells per space dimension.  The paper contains a wide set of test cases for different hyperbolic PDEsystems, solved on adaptive Cartesian meshes (AMR) that show the capabilitiesof the proposed method both on smooth and discontinuous problems, as well asthe broad range of its applicability. The tests range from compressiblegasdynamics over classical MHD to relativistic magnetohydrodynamics.

A posteriori subcell finite volume limiter for general PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics