A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

Abstract

This is the accepted version of the following article: [Huerta, A., Casoni, E. and Peraire, J. (2012), A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int. J. Numer. Meth. Fluids, 69: 1614–1632. doi:10.1002/fld.2654], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.2654/fullThis article presents a novel shock-capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high-order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high-order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach.Peer ReviewedPostprint (author's final draft

Solving 3D relativistic hydrodynamical problems with weighted essentially nonoscillatory discontinuous Galerkin methods

“…However, this allows us to isolate the effects due to the subcell algorithm and to establish its usefulness even in the absence of the other ingredients. There are open questions regarding the stability and accuracy of the subgrid method, in particular when used without the locally implicit time integration method (but see also [37,38]). …”

“…Another important development is a hybrid approach [35][36][37][38], which replaces troubled cells by an equidistant subgrid and applies FV shock capturing on these grids. This approach maintains the subcell resolution of FV methods, but increases the complexity of the implementation since two types of grids and special grid transfer operators are required.…”

A moving discontinuous Galerkin finite element method for flows with interfaces

“…However, this requires that the interfaces are aligned with the grid. When misaligned, instabilities can arise, leading to the common approach of artificial stabilization or shock capturing . While shock capturing has been remarkably successful for simulating a variety of complex flows, it can diminish some of the benefits of DG, including high‐order accuracy and consistency, which remains an ongoing issue as emphasized in a recent survey of higher‐order methods in the work of Wang et al…”

A stabilized Powell–Sabin finite-element method for the 2D Euler equations in supersonic regime

“…Most of the works focus on finite volumes (WENO techniques [3]) or discontinuous 5 Galerkin (DG) methods [4,5,6,7]. However in many physical applications, such as magnetohydrodynamics, shells analysis and vibrations, numerical methods with high degrees of regularity are of interest.…”