Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable / entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

show abstract

“…A sufficient condition for entropy stability of the semi-discrete problem ( 7) is given by (cf. [8,27])…”

Section: Property-preserving Flux Correctionmentioning

confidence: 99%

“…where ν e ij ≥ 0 is an entropy viscosity coefficient. This simple comparison principle provides a powerful tool for the design of entropy stable finite volume [11,29,34] and DG [8,27] methods. For our AFC scheme (7) to be entropy conservative, the fluxes g e,EC ij = −g e,EC ji would need to satisfy…”

Section: Entropy Stable Afc Schemesmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

View full text Add to dashboard Cite

show abstract

“…So far, we discussed semi-discrete DGSEM-LGL variants with tensor product expansions on possible curvilinear unstructured hexahedral meshes. Direct extensions of this variant include nonconforming meshes [166,167], moving meshes [168][169][170], different related versions such as e.g., the line DG method [171], and a fully discrete space-time approach without the assumption on time continuity [172][173][174][175]. An exciting recent development are explicit modified Runge-Kutta methods that retain the semi-discrete entropy stability estimates [176].…”

Section: Where To Go Next?mentioning

confidence: 99%

A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

Gassner

Winters

2021

Front. Phys.

View full text Add to dashboard Cite

In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.

show abstract

“…Because of this, the framework is readily extended to other high-order numerical methods that feature the SBP property, e.g. multi-block finite difference methods (Hicken et al, 2016;Crean et al, 2018) or alternative DG approaches (Pazner and Persson, 2019). Additionally, the split forms have been extended to many other systems of PDEs including:…”

Section: Epiloguementioning

confidence: 99%

Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations

Winters¹,

Kopriva²,

Gassner³

et al. 2020

Preprint

View full text Add to dashboard Cite

Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Concerning robustness, DG has undergone an extensive transformation over the past seven years into its modern form that provides statements on solution boundedness for linear and nonlinear problems.This chapter takes a constructive approach to introduce a modern incarnation of the DG spectral element method for the compressible Navier-Stokes equations in a three-dimensional curvilinear context. The groundwork of the numerical scheme comes from classic principles of spectral methods including polynomial approximations and Gauss-type quadratures. We identify aliasing as one underlying cause of the robustness issues for classical DG spectral methods. Removing said aliasing errors requires a particular differentiation matrix and careful discretization of the advective flux terms in the governing equations.

show abstract

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

Cited by 26 publications

References 45 publications

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations

Contact Info

Product

Resources

About