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High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Travis C. Fisher, Mark H. Carpenter
2013 Journal of Computational Physics  
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

Constantine Dafermos, Dietmar Kröner, Randall LeVeque
2008 Oberwolfach Reports  
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]

Ge Yan, Sharanjeet Kaur, Jeffery W. Banks, Jason E. Hicken
2021 arXiv   pre-print
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

Khosro Shahbazi, Nathan Albin, Oscar P. Bruno, Jan S. Hesthaven
2011 Journal of Computational Physics  
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

Jean-Luc Guermond, Bojan Popov, Laura Saavedra, Yong Yang
2017 SIAM Journal on Scientific Computing  
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

Len G. Margolin, Roy S. Baty
2019 Journal of Geometric Mechanics (JGM)  
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

Khosro Shahbazi, Jan S. Hesthaven, Xueyu Zhu
2013 Journal of Computational Physics  
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]

Dmitri Kuzmin, Manuel Quezada de Luna
2020 arXiv   pre-print
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

Ulrik S. Fjordholm, Siddhartha Mishra, Eitan Tadmor
2012 SIAM Journal on Numerical Analysis  
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

Andreas Hiltebrand, Siddhartha Mishra
2013 Numerische Mathematik  
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

Jean-Luc Guermond, Richard Pasquetti, Bojan Popov
2011 Journal of Computational Physics  
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]

Dmitri Kuzmin, Manuel Quezada de Luna
2020 arXiv   pre-print
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  
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

Neelabja Chatterjee, Ulrik Skre Fjordholm
2019 Mathematical Modelling and Numerical Analysis  
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

Andreas Hiltebrand, Siddhartha Mishra
2015 Communications in Computational Physics  
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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