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High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains
2013
Journal of Computational Physics
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Conclusions High-order entropy stable finite difference methods have been derived for conservation laws on finite domains. ...
In this work, we have constructed high-order entropy stable finite difference schemes for finite domains by first developing a formal set of conditions based on a generalized summation-by-parts property ...
The second-to-last term on
Technical Memorandum
TITLE AND SUBTITLE High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains 5a. ...
doi:10.1016/j.jcp.2013.06.014
fatcat:q6nkn7dwebfw7oarbkrhlzyrgy
Hyperbolic Conservation Laws
2008
Oberwolfach Reports
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Discrete Entropy Inequalities in MHD Tim Barth Discrete entropy inequalities for the discontinuous Galerkin (DG) finite element discretization [12, 6, 5, 4] of first-order systems of nonlinear conservation ...
The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic-hyperbolic type for the velocity ...
Across shocks this can lead to a divergence error that is O(1); this in turn can
Entropy Stability Theory for Nonlinear Conservation Laws Eitan Tadmor Abstract. ...
doi:10.4171/owr/2008/56
fatcat:vwgkrt2egzgr3krlmfhngtxm4i
Entropy-stable discontinuous Galerkin difference methods for hyperbolic conservation laws
[article]
2021
arXiv
pre-print
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The paper describes the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. ...
for other entropy-stable discretizations. ...
The need for robust and efficient high-order discretizations motivates this work on entropy-stable discontinuous Galerkin difference methods. ...
arXiv:2103.03826v1
fatcat:2mep5qsb3va2hntvwaz6rjgafa
Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws
2011
Journal of Computational Physics
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We introduce a multi-domain Fourier-Continuation/WENO hybrid method (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially ...
We consider WENO schemes of orders five and nine and a FC method of order five. ...
WENO Methods for conservation laws with discontinuous solutions WENO finite difference methods are finite difference schemes tailored to conservation laws involving discontinuous solutions-or, more generally ...
doi:10.1016/j.jcp.2011.08.024
fatcat:czocl7lw5vgk3i5hl2znnluyeu
Invariant Domains Preserving Arbitrary Lagrangian Eulerian Approximation of Hyperbolic Systems with Continuous Finite Elements
2017
SIAM Journal on Scientific Computing
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A conservative invariant domain preserving arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems is introduced. ...
element literature, and leads to precise invariant domain properties and entropy inequalities; (v) the methods works for any (reasonable in the sense of section 2) hyperbolic system. ...
high-order finite elements have good properties in terms of geometry representation, symmetry preservation, resolution of shock fronts, and high-order convergence rate for smooth solutions. ...
doi:10.1137/16m1063034
fatcat:4k3wvqhvwjeljoiumhwedxy4sm
Conservation laws in discrete geometry
2019
Journal of Geometric Mechanics (JGM)
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In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. ...
The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry ...
It is our purpose to show how this limiting process can be avoided for the purpose of finding entropy stable numerical algorithms for finite volume schemes.
Kinetic theory. ...
doi:10.3934/jgm.2019010
fatcat:ujrihgi765birhghbcfgd3ptli
Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws
2013
Journal of Computational Physics
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We introduce a multi-dimensional point-wise multi-domain hybrid Fourier-Continuation/WENO technique (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws ...
difference-WENO (CD-WENO) schemes. ...
WENO Methods for conservation laws with discontinuous solutions WENO finite difference methods are finite difference schemes tailored to enable the accurate solution of conservation laws involving discontinuous ...
doi:10.1016/j.jcp.2013.07.009
fatcat:zvowm6irsra5dda5xaean5zyx4
Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws
[article]
2020
arXiv
pre-print
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Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems. ...
The new methodology combines the advantages of modern entropy stable/entropy conservative schemes and their local extremum diminishing counterparts. ...
The authors would like to thank Hennes Hajduk (TU Dortmund University) for careful proofreading of the manuscript and helpful feedback. ...
arXiv:2003.12007v1
fatcat:izd5q2h7ynhjveavnj7sro4zom
Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws
2012
SIAM Journal on Numerical Analysis
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We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. ...
Secondorder entropy stable schemes for scalar conservation laws were presented in [25] . ...
We will design arbitrary-order accurate finite difference schemes that satisfy a discrete version of (6.3).
Entropy stable finite difference schemes. ...
doi:10.1137/110836961
fatcat:hnkqd7svozdq5kydruvsbopqvq
Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws
2013
Numerische Mathematik
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We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. ...
Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. ...
Barth, NASA, Ames, USA for interesting discussions on spacetime DG methods and for providing the particular forms of the shock capturing operators used by him. ...
doi:10.1007/s00211-013-0558-0
fatcat:4l7ywznnijf3lhh26mifdyf33y
Entropy viscosity method for nonlinear conservation laws
2011
Journal of Computational Physics
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A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). ...
The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. ...
We describe in this paper a new way of generating high-order numerical approximations for nonlinear conservation laws. ...
doi:10.1016/j.jcp.2010.11.043
fatcat:vvpvhhrjsbfbddsxh6cgzoiue4
Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws
[article]
2020
arXiv
pre-print
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This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. ...
We prove that the standard CG method is entropy conservative for the square entropy. ...
The authors would like to thank Hennes Hajduk (TU Dortmund University) for suggesting an improved version of the subcell flux decomposition. ...
arXiv:2005.08788v1
fatcat:p5o3rr3nd5fvranrtbneamfpmm
1. Hyperbolic Conservation Laws
[chapter]
2000
Hyperbolic and Viscous Conservation Laws
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We consider approximate solutions to nonlinear hyperbolic conservation laws. ...
Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. ...
Acknowledgements The authors are grateful to Rémi Abgrall for helpful early discussions, and to Bernardo Cockburn for eye-opening conversations. The work of S. ...
doi:10.1137/1.9780898719420.ch1
fatcat:2c6bxcjrajhbxem6axchrzxat4
Convergence of second-order, entropy stable methods for multi-dimensional conservation laws
2019
Mathematical Modelling and Numerical Analysis
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High-order accurate, \emph{entropy stable} numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available ...
entropy condition. ...
The authors would like to thank SAM for the warm hospitality and facilities, and Prof. Dr. Siddhartha Mishra for discussing the problem. N.C.' ...
doi:10.1051/m2an/2019090
fatcat:dpukmndbyzbuldkx3zehazhcmi
Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws
2015
Communications in Computational Physics
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AbstractAn entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper to approximate hyperbolic systems of conservation laws. ...
This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. ...
More recently, arbitrarily high-order entropy stable finite difference schemes were proposed in [10] . ...
doi:10.4208/cicp.140214.271114a
fatcat:4koxwpmezza6padarjcrsy6vjy
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