Book ChapterDOI
Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Subcells
Matthias Sonntag,Claus-Dieter Munz +1 more
- pp 945-953
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TLDR
This approach combines the good properties of the discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations.Abstract:
We present a shock capturing procedure for high order discontinuous Galerkin methods, by which shock regions are refined and treated by the finite volume techniques. Hence, our approach combines the good properties of the discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the subcell approach the interior resolution on the discontinuous Galerkin grid cell is preserved and the number of degrees of freedom remains the same. In this paper we focus on an implementation of this coupled method and show our first results.read more
Citations
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Journal ArticleDOI
A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws
TL;DR: A novel a posteriori finite volume subcell limiter technique for the Discontinuous Galerkin finite element method for nonlinear systems of hyperbolic conservation laws in multiple space dimensions that works well for arbitrary high order of accuracy in space and time and that does not destroy the natural subcell resolution properties of the DG method.
Journal ArticleDOI
High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics
TL;DR: The direct connection between the HPR model and the classical hyperbolic-parabolic Navier-Stokes-Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit.
Journal ArticleDOI
Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting
TL;DR: A novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space–time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order a posteriori sub-cell ADER-WENO finite volume limiter.
Journal ArticleDOI
A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes
Michael Dumbser,Raphaël Loubère +1 more
TL;DR: This novel a posteriori limiter, which has been recently proposed for the simple Cartesian grid case in 62, is able to resolve discontinuities at a sub-grid scale and is substantially extended here to general unstructured simplex meshes in 2D and 3D.
Journal ArticleDOI
Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
TL;DR: In this paper, a new numerical tool for solving the relativistic ideal MHD equations is presented, based on the combination of the following three key features: (i) a one-step ADER discontinuous Galerkin (DG) scheme that allows for an arbitrary order of accuracy in both space and time, (ii) an a posteriori subcell finite volume limiter that is activated to avoid spurious oscillations at discontinuities without destroying the natural subcell resolution capabilities of the DG finite element framework and finally (iii) a space-time adaptive mesh refinement
References
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Journal ArticleDOI
The numerical simulation of two-dimensional fluid flow with strong shocks
Paul R. Woodward,Phillip Colella +1 more
TL;DR: In this paper, a comparison of numerical methods for simulating hydrodynamics with strong shocks in two dimensions is presented and discussed, and three approaches to treating discontinuities in the flow are discussed.
Journal ArticleDOI
High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws
TL;DR: The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is described in this article.
Journal ArticleDOI
A Method for the Numerical Calculation of Hydrodynamic Shocks
TL;DR: In this paper, the equations of hydrodynamics are modified by the inclusion of additional terms which greatly simplify the procedures needed for stepwise numerical solution of the equations in problems involving shocks.
Proceedings ArticleDOI
Sub-Cell Shock Capturing for Discontinuous Galerkin Methods
Per-Olof Persson,Jaime Peraire +1 more
TL;DR: A shock capturing strategy for higher order Discontinuous Galerkin approximations of scalar conservation laws is presented and it is shown that the proposed approach is capable of capturing the shock as a sharp, but smooth profile, which is typically contained within one element.
Journal ArticleDOI
A discontinuous hp finite element method for the Euler and Navier–Stokes equations
TL;DR: A new method for the solution of the Euler and Navier-Stokes equations is introduced, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver.