Abstract
In this work, we consider the general family of the so called ADER P_NP_M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step P_NP_M schemes 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), the usual Discontinuous Galerkin (DG) methods (N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with M>N. In all cases with M ge N > 0 the P_NP_M schemes are linear in 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 schemes. 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 schemes, due to the use of a rather fine subgrid of 2N+1 subcells 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.
Highlights
In this work we want to improve the family of high order accurate ADER PN PM schemes first introduced in [37,40] for the solution of hyperbolic partial differential equations
The reconstruction procedure is divided into two steps: concerning the spatial reconstruction, we employ a classical WENO reconstruction in the case of pure finite volume schemes (N = 0), a reconstruction procedure based on L2 projection that is linear in the sense of Godunov for N > 0 and M > N, and in the case or pure Discontinous Galerkin (DG) schemes (N = M) the reconstruction reduces to the identity operator; concerning the reconstruction in time, we employ a novel variant of the ADER approach of Toro and Titarev, see [26,109,110,113,114], based on an element-local space-time Galerkin predictor, see [40]
Examples of hyperbolic equations are the Euler equations of gasdynamics, the Shallow Water equations [27,108] and many multiphase models [4,38,62] used in fluid mechanics, the magnetohydrodynamics system (MHD) for plasma flow [6,9], the unified first order hyperbolic formulation of continuum mechanics by Godunov, Peshkov and Romenski (GPR) [43,48,49,65,66,93] as well as the special and general relativistic formulations of MHD, see e.g. [3,5,10,33,41,56,122], or for the Einstein field equations (CCZ4) [1,2,42,44]
Summary
In this work we want to improve the family of high order accurate ADER PN PM schemes first introduced in [37,40] for the solution of hyperbolic partial differential equations In this family of schemes the discrete solution is represented in space through high order piecewise polynomials of degree N at each timestep; the data are evolved in time through a spacetime reconstruction procedure of order M. We can see the Finite Volume (FV) schemes of order M as a particular case of PN PM methods when N = 0, and the Discontinous Galerkin (DG) methods are included in this family when choosing N = M This family contains another important class of hybrid or reconstructed DG schemes when taking N > 0, M > N , which are the main object of study of this paper. The nominal order of accuracy of the scheme is given by M + 1 so it can be at least in principle arbitrary high
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