A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

Abstract

AbstractIn this work, we consider the general family of the so called ADER$$P_NP_M$$PNPMschemes for the numerical solution of hyperbolic partial differential equations witharbitraryhigh order of accuracy in space and time. The family of one-step$$P_NP_M$$PNPMschemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$N=0), the usual Discontinuous Galerkin (DG) methods ($$N=M$$N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degreeMis applied over piecewise polynomial data of degreeNwith$$M>N$$M>N. In all cases with$$M \ge N > 0 $$M≥N>0the$$P_NP_M$$PNPMschemes arelinearin the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of$$P_NP_M$$PNPMschemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order$$P_NP_M$$PNPMschemes, due to the use of a rather fine subgrid of$$2N+1$$2N+1subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.