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In this paper we present a novel arbitrary high order accurate discontinuousGalerkin (DG) finite element method on space-time adaptive Cartesian meshes(AMR) for hyperbolic conservation laws in multiple space dimensions, using ahigh order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}.Notoriously, the original DG method produces strong oscillations in thepresence of discontinuous solutions and several types of limiters have beenintroduced over the years to cope with this problem. Following the innovativeidea recently proposed in \cite{Dumbser2014}, the discrete solution within thetroubled cells is \textit{recomputed} by scattering the DG polynomial at theprevious time step onto a suitable number of sub-cells along each direction.Relying on the robustness of classical finite volume WENO schemes, the sub-cellaverages are recomputed and then gathered back into the DG polynomials over themain grid. In this paper this approach is implemented for the first time withina space-time adaptive AMR framework in two and three space dimensions, afterassuring the proper averaging and projection between sub-cells that belong todifferent levels of refinement. The combination of the sub-cell resolution withthe advantages of AMR allows for an unprecedented ability in resolving even thefinest details in the dynamics of the fluid. The spectacular resolutionproperties of the new scheme have been shown through a wide number of testcases performed in two and in three space dimensions, both for the Eulerequations of compressible gas dynamics and for the magnetohydrodynamics (MHD)equations.

Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting