A p-Adaptive Discontinuous Galerkin Method with hp-Shock Capturing

Abstract

In this work, we present a novel hybrid Discontinuous Galerkin scheme with hp-adaptivity capabilities for the compressible Euler equations. In smooth regions, an efficient and accurate discretization is achieved via local p-adaptation. At strong discontinuities and shocks, a finite volume scheme on an h-refined element-local subgrid gives robustness. Thus, we obtain a hp-adaptive scheme that exploits both the high convergence rate and efficiency of a p-adaptive high order scheme as well as the stable and accurate shock capturing abilities of a low order finite volume scheme, but avoids the inherent resolution loss through h-refinement. A single a priori indicator, based on the modal decay of the local polynomial solution representation, is used to distinguish between discontinuous and smooth regions and control the p-refinement. Our method is implemented as an extension to the open source software FLEXI. Hence, the efficient implementation of the method for high performance computers was an important criterion during the development. The efficiency of our adaptive scheme is demonstrated for a variety of test cases, where results are compared against non adaptive simulations. Our findings suggest that the proposed adaptive method produces comparable or even better results with significantly less computational costs.

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

“…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]. While referring to the cited literature for more details, and in particular to [80] for the complete description of our limiter on moving polygonal meshes, here we just briefly recall the key passages and illustrate the small details necessary to make the resulting scheme well-balanced.…”

An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows

“…The indirect (output‐based) error estimators estimates the error of a targeted quantity indirectly usually through solving an adjoint equation [1, 14, 20, 21, 27]. The direct (solution‐based) error estimators or indicators are mainly divided into three types, (a) residual‐based estimators [6, 15, 21, 23], which evaluate the local residual term inside cells and at cell faces; (b) gradient‐based or interface jump‐based indicators, which detect complex structures (such as shock waves, vortices) through gradients of physical quantities (such as gradient of density [33] and pressure [4]) or interface jumps of DG solutions [3, 11, 34]; (c) recovery‐based estimators [30, 41, 44, 45], which evaluate the regularity of the solution by computing the distance between original numerical solution and an enhanced solution $$$ {\left\Vert {u}_h-{u}^{\ast}\right\Vert}_{L_2} $$$, where $$$ {u}^{\ast } $$$ denotes an enhanced solution through order enrichment or mesh refinement.…”