A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Abstract

In this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

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 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].…”

An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing

“…For readability, in the remaining parts of this paper, we analyze and discretize the one-dimensional GLM-MHD equations, (23). This can be done without loss of generality, as the spatial dimensions are decoupled in the GLM-MHD system [14].…”

“…Bohm et al [8] proposed an entropy stable DGSEM discretization of the resistive GLM-MHD equations. To obtain it, we rewrite (23) as…”

“…This approach was later extended to MHD by Núñez-de la Rosa and Munz [22]. More general approaches have been suggested by Markert et al [23], who proposed a continuous blending between DG schemes of different orders and a subcell FV method; and by Vilar [24], who presented a so-called a posteriori limitation procedure for DG that uses an underlying subcell FV scheme to control the monotonicity and positivity of the numerical solution. More recently, subcell FV shock-capturing methods that satisfy entropy inequalities have been presented [25,26,27].…”