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 the so called ADER $$P_NP_M$$ P N P 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 $$P_NP_M$$ P N P 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 ($$N=0$$ 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 degree M is applied over piecewise polynomial data of degree N with $$M>N$$ M > N . In all cases with $$M \ge N > 0 $$ M ≥ N > 0 the $$P_NP_M$$ P N P 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 $$P_NP_M$$ P N P 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 $$P_NP_M$$ P N P M schemes, due to the use of a rather fine subgrid of $$2N+1$$ 2 N + 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.

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

“…In the case of shock waves and other discontinuities that might occur even when starting the simulation with smooth initial data, because of the non-linearity of the governing PDE system, a limiter technique must be introduced in the DG scheme (3.30) in order to avoid spurious oscillations and dangerous instabilities. Among the variety of techniques presented in literature [14,17,35,44,47,51,67,69,79,96,97] that range from classical a priori limiters to novel a posteriori limiting techniques, and that exploit the combination of different schemes, bounds preserving approaches and/or different level of refinements, here we have chosen to introduce a simple artificial viscosity method, inspired from [7,55,65,70,73,84], that we apply only to those cells that need limitation, i.e. those which are detected as "troubled".…”

High-order Arbitrary-Lagrangian-Eulerian schemes on crazy moving Voronoi meshes

“…The key idea of direct ALE methods (in contrast to indirect ones) consists in connecting two tessellations by means of so-called space-time control volumes 𝐶 𝑛 𝑖 , and recover the unknown solution at the new timestep u 𝑛+1 ℎ directly inside the new polygon 𝑃 𝑛+1 𝑖 , from the data available at the previous timestep u 𝑛 ℎ in 𝑃 𝑛 𝑖 . This is achieved through the integration, over such control volumes, of the fluxes, the nonconservative products and the source terms, by means of a high-order fully discrete predictor-corrector ADER method [21,31]. In this way, the need for any further remapping/remeshing steps is totally eliminated.…”

“…High-order schemes that can be seen as linear in the sense of Godunov [34], may develop spurious oscillations in presence of discontinuities. In order to prevent this phenomenon, in the case of a DG discretization we adopt an a posteriori limiting procedure based on the MOOD paradigm [15,47,31]: we first apply our unlimited ALE-DG scheme everywhere, and then (a posteriori), at the end of each timestep, we check the reliability of the obtained solution in each cell against physical and numerical admissibility criteria, such as floating point exceptions, violation of positivity or violation of a relaxed discrete maximum principle (and see [35,39] for further criteria). Next, we mark as troubled those cells where the DG solution cannot be accepted.…”

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

“…Among the different approaches available in literature, as those inspired to Cockburn and Shu [46,47], based on the use of a total variation bounded limiter, or the moment limiters [113], the artificial viscosity procedures [143] WENO-type limiters [149,150], or gradient-based limiters [118,116], we have selected the so-called a posteriori subcell finite volume (FV) limiter. This type of limiter is based on the MOOD approach [45,124], which has already been successfully applied in the ALE finite volume framework in [26,25] and in the discontinuous Galerkin case in [156,157,52,98,166,130,152,148] and, with a notation similar to the one used here, in [74,175,70,174,108,80,85]. We finally remark that shockcapturing techniques, based on subcell finite volume schemes, can also be applied in a predictive (a priori) fashion, for example as in [156,157,11,141,87].…”