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

Abstract In this work, we consider the general family of so called ADER $$P_NP_M$$ PNPM schemes for numerical solution hyperbolic partial differential equations with arbitrary high order accuracy in space and time. The one-step was introduced Dumbser (J Comput Phys 227:8209–8253, 2008) represents a unified framework classical Finite Volume (FV) ( $$N=0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">N=0 ), usual Discontinuous Galerkin (DG) methods $$N=M$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">N=M as well new class intermediate hybrid which reconstruction operator degree M is applied over piecewise polynomial data N $$M>N$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">M>N . all cases $$M \ge > 0 $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">M?N>0 are linear sense Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear even destroy simulation. Therefore, paper present simple, robust accurate posteriori subcell finite volume limiting strategy that valid entire schemes. FV limiter activated only where it needed, i.e. neighborhood shocks or other able to maintain resolution underlying schemes, due use rather fine subgrid $$2N+1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">2N+1 subcells per dimension. contains wide set test different PDE systems, solved on adaptive Cartesian meshes show capabilities proposed method both smooth discontinuous problems, broad range its applicability. tests from compressible gasdynamics MHD relativistic magnetohydrodynamics.