skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Abstract

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers’ equation and the Euler equations, in one, two, and three spatial dimensions.

Authors:
ORCiD logo [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Univ. of California, Berkeley, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1548369
Report Number(s):
LLNL-JRNL-767379
Journal ID: ISSN 0885-7474; 957725
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Scientific Computing
Additional Journal Information:
Journal Volume: 80; Journal Issue: 1; Journal ID: ISSN 0885-7474
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Entropy stability; Discontinuous Galerkin; Line-DG; Spectral element method

Citation Formats

Pazner, Will, and Persson, Per-Olof. Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method. United States: N. p., 2019. Web. doi:10.1007/s10915-019-00942-1.
Pazner, Will, & Persson, Per-Olof. Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method. United States. doi:10.1007/s10915-019-00942-1.
Pazner, Will, and Persson, Per-Olof. Wed . "Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method". United States. doi:10.1007/s10915-019-00942-1. https://www.osti.gov/servlets/purl/1548369.
@article{osti_1548369,
title = {Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method},
author = {Pazner, Will and Persson, Per-Olof},
abstractNote = {We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers’ equation and the Euler equations, in one, two, and three spatial dimensions.},
doi = {10.1007/s10915-019-00942-1},
journal = {Journal of Scientific Computing},
number = 1,
volume = 80,
place = {United States},
year = {2019},
month = {3}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Spectral element approximation of convection–diffusion type problems
journal, May 2000


Entropy Stable Spectral Collocation Schemes for the Navier--Stokes Equations: Discontinuous Interfaces
journal, January 2014

  • Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J.
  • SIAM Journal on Scientific Computing, Vol. 36, Issue 5
  • DOI: 10.1137/130932193

On discretely entropy conservative and entropy stable discontinuous Galerkin methods
journal, June 2018


Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for Compressible Euler and Navier-Stokes Equations
journal, November 2013


Inviscid and Viscous Simulations of the Taylor-Green Vortex Flow Using a Modal Discontinuous Galerkin Approach
conference, September 2012

  • Chapelier, Jean-Baptiste; De La Llave Plata, Marta; Renac, Florent
  • 42nd AIAA Fluid Dynamics Conference and Exhibit
  • DOI: 10.2514/6.2012-3073

Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws
journal, September 2017


The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws
journal, January 1991

  • Cockburn, Bernardo; Shu, Chi-Wang
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 25, Issue 3
  • DOI: 10.1051/m2an/1991250303371

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains
journal, November 2013


A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods
journal, January 2013

  • Gassner, Gregor J.
  • SIAM Journal on Scientific Computing, Vol. 35, Issue 3
  • DOI: 10.1137/120890144

Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations
journal, December 2016

  • Gassner, Gregor J.; Winters, Andrew R.; Kopriva, David A.
  • Journal of Computational Physics, Vol. 327
  • DOI: 10.1016/j.jcp.2016.09.013

On the symmetric form of systems of conservation laws with entropy
journal, January 1983


Solutions of Multi-dimensional Hyperbolic Systems of Conservation Laws by Square Entropy Condition Satisfying Discontinuous Galerkin Method
journal, September 2006


A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics
journal, February 1986

  • Hughes, T. J. R.; Franca, L. P.; Mallet, M.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 54, Issue 2
  • DOI: 10.1016/0045-7825(86)90127-1

Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks
journal, August 2009


On a Cell Entropy Inequality for Discontinuous Galerkin Methods
journal, April 1994

  • Jiang, Guangshan; Shu, Chi-Wang
  • Mathematics of Computation, Vol. 62, Issue 206
  • DOI: 10.2307/2153521

Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes
journal, March 2006


A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows
journal, April 1996

  • Kopriva, David A.; Kolias, John H.
  • Journal of Computational Physics, Vol. 125, Issue 1
  • DOI: 10.1006/jcph.1996.0091

On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence
journal, February 2017


Spectral methods for problems in complex geometries
journal, August 1980


On the convergence of difference approximations to scalar conservation laws
journal, January 1988


Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations
journal, January 2016

  • Parsani, Matteo; Carpenter, Mark H.; Fisher, Travis C.
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 5
  • DOI: 10.1137/15m1043510

High-Order DNS and LES Simulations Using an Implicit Tensor-Product Discontinuous Galerkin Method
conference, June 2017

  • Pazner, Will; Persson, Per-Olof
  • 23rd AIAA Computational Fluid Dynamics Conference
  • DOI: 10.2514/6.2017-3948

Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations
journal, April 2017


Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
journal, February 2018


High-Order Navier-Stokes Simulations Using a Sparse Line-Based Discontinuous Galerkin Method
conference, November 2012

  • Persson, Per-Olof
  • 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
  • DOI: 10.2514/6.2012-456

A sparse and high-order accurate line-based discontinuous Galerkin method for unstructured meshes
journal, January 2013


Sub-Cell Shock Capturing for Discontinuous Galerkin Methods
conference, June 2012

  • Persson, Per-Olof; Peraire, Jaime
  • 44th AIAA Aerospace Sciences Meeting and Exhibit
  • DOI: 10.2514/6.2006-112

Comparison of Some Entropy Conservative Numerical Fluxes for the Euler Equations
journal, December 2017


An Efficient Implicit Discontinuous Spectral Galerkin Method
journal, September 2001

  • Rasetarinera, Patrick; Hussaini, M. Y.
  • Journal of Computational Physics, Vol. 172, Issue 2
  • DOI: 10.1006/jcph.2001.6853

Numerical Convergence Study of Nearly Incompressible, Inviscid Taylor–Green Vortex Flow
journal, July 2005

  • Shu, Chi-Wang; Don, Wai-Sun; Gottlieb, David
  • Journal of Scientific Computing, Vol. 24, Issue 1
  • DOI: 10.1007/s10915-004-5407-y

Auto-tuning of level 1 and level 2 BLAS for GPUs: AUTO-TUNING OF LEVEL 1 AND LEVEL 2 BLAS FOR GPUs
journal, September 2012

  • Sørensen, Hans Henrik Brandenborg
  • Concurrency and Computation: Practice and Experience, Vol. 25, Issue 8
  • DOI: 10.1002/cpe.2916

The numerical viscosity of entropy stable schemes for systems of conservation laws. I
journal, September 1987


Geometric Conservation Law and Its Application to Flow Computations on Moving Grids
journal, October 1979

  • Thomas, P. D.; Lombard, C. K.
  • AIAA Journal, Vol. 17, Issue 10
  • DOI: 10.2514/3.61273

From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations
journal, July 2010

  • Vos, Peter E. J.; Sherwin, Spencer J.; Kirby, Robert M.
  • Journal of Computational Physics, Vol. 229, Issue 13
  • DOI: 10.1016/j.jcp.2010.03.031

High-order CFD methods: current status and perspective: HIGH-ORDER CFD METHODS
journal, January 2013

  • Wang, Z. J.; Fidkowski, Krzysztof; Abgrall, Rémi
  • International Journal for Numerical Methods in Fluids, Vol. 72, Issue 8
  • DOI: 10.1002/fld.3767

A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations
journal, November 2018

  • Winters, Andrew R.; Moura, Rodrigo C.; Mengaldo, Gianmarco
  • Journal of Computational Physics, Vol. 372
  • DOI: 10.1016/j.jcp.2018.06.016

Performance tuning of Newton-GMRES methods for discontinuous Galerkin discretizations of the Navier-Stokes equations
conference, June 2013

  • Zahr, Matthew J.; Persson, Per-Olof
  • 21st AIAA Computational Fluid Dynamics Conference
  • DOI: 10.2514/6.2013-2685

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes
journal, November 2010


Implementation of the entropy viscosity method with the discontinuous Galerkin method
journal, January 2013

  • Zingan, Valentin; Guermond, Jean-Luc; Morel, Jim
  • Computer Methods in Applied Mechanics and Engineering, Vol. 253
  • DOI: 10.1016/j.cma.2012.08.018