ENTROPY STABILITY

A collection of selected references on

Entropy conservative and entropy stable approximations
of nonlinear conservation laws and related problems




  • E. Tadmor
    Skew self-adjoint form for systems of conservation laws
    Journal of Mathematical Analysis and Applications 103(2) (1984) 428-442.

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  • E. Tadmor
    Numerical viscosity and the entropy condition for conservative difference schemes
    Mathematics of Computation 43 (1984) 369-381.

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  • E. Tadmor
    A minimum entropy principle in the gas dynamics equations
    Applied Numerical Mathematics 2 (1986) 211-219.

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  • E. Tadmor
    Entropy conservative finite element schemes
    in "Numerical Methods for Compressible Flows - Finite Difference Element and Volume Techniques", Proceedings of the winter annual meeting of the American Society of Mechanical Engineering, Vol. AMD-Vol. 78 (T. E. Tezduyar and T.J.R. Hughes, eds.) (1986) 149-158..

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  • E. Tadmor
    Entropy functions for symmetric systems of conservation laws
    Journal of Mathematical Analysis and Applications 122(2) (1987) 355-359.

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  • E. Tadmor
    The entropy dissipation by numerical viscosity in nonlinear conservative difference schemes
    in "Nonlinear Hyperbolic Problems'", Proceedings of a 1986 Advanced Research Workshop in St. Etienne, Lecture Notes in Mathematics, Vol. Vol. 1270 (C. Carasso, P.-A. Raviart and D. Serre, eds.), Springer-Verlag (1987) 52-63..

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  • E. Tadmor
    The numerical viscosity of entropy stable schemes for systems of conservation laws. I.
    Mathematics of Computation 49 (1987) 91-103.

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  • M. Merriam
    An entropy based approach to nonlinear stability
    NASA Technical Memorandum 101086 (1989).

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  • F. Dubois
    Concavity of thermostatic entropy and convexity of LAX's mathematical entropy
    La Recherche Aerospatiale (English Edition) (ISSN 0379-380X) 3 (1990) 77-80.

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  • F. Bouchut, Ch. Bourdarias and B. Perthame
    A MUSCL method satisfying all the numerical entropy inequalities
    Mathematics of Computation 65 (1996) 65 (1996), 1439-1461.

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  • B. Hayes and P. LeFloch
    Nonclassical shocks and kinetic relations: finite difference schemes
    SIAM J. Numerical Analysis 35(6) (1998) 2169-2194.

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  • C. Rhode and P. LeFloch
    High-order schemes, entropy inequalities, and nonclassical shocks
    SIAM J. Numerical Analysis 37(6) (2000) 2023-2060.

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  • C. Chalons and P. LeFloch
    High-order entropy-conservative schemes and kinetic relations for van der Waals fluids
    Journal of Computational Physics 168(1) (2001) 184-206.

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  • C. Chalons and P. LeFloch
    A fully discrete scheme for diffusive-dispersive conservation laws
    Numerische Mathematik 89(3) (2001) 493-509.

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  • C. Chalons and P. G. LeFloch
    High-order entropy-conservative schemes and kinetic relations for van der Waals fluids
    Journal of Computational Physics 168(1) (2001) 184-2006.

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  • P. LeFloch, J.M. Mercier and C. Rohde
    Fully discrete, entropy conservative schemes of arbitrary order
    SIAM J. Numerical Analysis 40(5) (2002) 1968-1992.

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  • E. Tadmor
    Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems
    Acta Numerica 12 (2003) 451-512.

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  • R. Abramov and J. Majda
    Discrete approximations with additional conserved quantities: deterministic and statistical behavior
    Methods and Applications in Analysis 10(2) (2003) 151-189.

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  • G. F. Naterer and J. A. Camberos
    Entropy and the second law fluid flow and heat transfer simulation
    AIAA Journal of Thermophysics and Heat Transfer 17(3) (2003) 360-371.

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  • E. Tadmor and W. Zhong
    Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity
    J. of Hyperbolic Differential Equations 3(3) (2006) 529-559.

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  • E. Tadmor
    On the entropy stability of difference schemes: a comparison principle and a homotopy approach
    in "Hyperbolic Problems: Theory, Numerics, Applications'", Proceedings of the 10th International Conference in 2004, Vol. vol. I (F. Asukura, H. Aiso, S. Kawashima, A. Matsumura, S. Nishibata & K. Nishihara, eds.), Yokohama Publishers, Osaka (2006) 195-204..

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  • P. Roe
    Affordable, entropy-consistent, Euler flux functions (with applications to the carbuncle phenomenon)
    in "11th International Conference on `Hyperbolic Problems: Theory, Numerics, Applications'", Lyon (2006).

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  • E. Tadmor and W. Zhong
    Novel entropy stable schemes for 1D and 2D fluid equations
    in "Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 11th International Conference in Lyon in 2006 (S. Benzoni-Gavage and D. Serre, eds.), Springer (2007) 1111-1120..

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  • J. Haink and C. Rohde
    Local discontinuous-Galerkin schemes for model problems in phase transition theory
    Communications in Computational Physics 4(4) (2008) 860-893.

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  • P. LeFloch and M. Mohammadian
    Why many theories of shock waves are necessary: kinetic functions, equivalent equations, and fourth-order models
    J. Computational Physics 227(8) (2008) 4162-4189.

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  • D. Kroner, P. LeFloch and M.-D. Thanh
    The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section
    Mathematical Modeling & Numerical Analysis 42(3) (2008) 425-442.

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  • E. Tadmor and W. Zhong
    Energy-preserving and stable approximations for the two-dimensional shallow water equations
    in "Mathematics and Computation - A Contemporary View", Proceedings of the Third Abel Symposium held in Ĺlesund, Norway in 2006 (H. Munthe-Kaas & B. Owren eds.), Springer (2008) 67-94..

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  • Jean-Luc Guermond and Richard Pasquetti
    Entropy-based nonlinear viscosity for Fourier approximations of conservation laws
    C. R. Acad. Sci. Paris, Ser. I 346 (2008) 801-806.

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  • J. M. Mercier
    Optimally transported schemes: one dimensional case
    (2008).

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  • P. LeFloch and S. Mishra
    Kinetic functions in magnetohydrodynamics with resistivity and Hall effect
    Acta Mathematica Scientia 29B(6) (2009) 1684-1702.

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  • K. Kitamura, P. Roe and F. Ismail
    Evaluation of Euler fluxes for hypersomic flow computations
    AIAA 47(1) (2009) 44-53.

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  • M. Lukacova- Medvidova and K. W. Morton
    Finite volume evolution Galerkin methods: A survey
    Indian J. Pure & Appl. Math. (2009).

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  • B. Sjogreen and H. Yee
    On skew-symmetric splitting and entropy conservation for the Euler Equations
    in "Numerical Mathematics and Advanced Applications", Proc of ENUMATH2009 8th European Conference held in Uppsala in 2009 (G. Kreiss, P. Lotstedt, A. Malquist, M. Neytcheva eds.), Springer (2009) 817-827..

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  • U. Fjordholm, S. Mishra and E. Tadmor
    Energy preserving and energy stable schemes for the shallow water equations
    in "Foundations of Computational Mathematics", Proc. FoCM in 2008 (F. Cucker, A. Pinkus & M. Todd, eds), London Math. Soc. Lecture Notes Ser. 363, Hong Kong (2009) 93-139..

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  • M. Lukacova- Medvidova and E. Tadmor
    On the entropy stability of Roe-type finite volume methods
    in "Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 12th International Conference in 2008, Vol. 67(2) (E. Tadmor, J.-G. Liu & A. Tzavaras, eds.), AMS Proc. Symp. Applied Math., University of Maryland (2009) 765-774..

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  • Aziz Madrane and Eitan Tadmor
    Entropy stability of Roe-type upwind finite volume methods on unstructured grids
    in "Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 12th International Conference in 2008, Vol. 67(2) (E. Tadmor, J.-G. Liu & A. Tzavaras, eds.), AMS Proc. Symp. Applied Math., University of Maryland (2009) 775-784..

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  • Susana Serna
    A characteristic-based nonconvex entropy fix upwind scheme for the ideal magnetohydrodynamic equations
    Journal of Computational Physics 228 (2009) 4232-4247.

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  • F. Ismail and P. L. Roe
    Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks
    J. Computational Physics 228 (2009) 5410-5436.

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  • Christos Arvanitis, Charalambos Makridakis and Nikolaos Sfakianakis
    Entropy conservative schemes and adapive mesh selection for hyperbolic conservation laws
    Journal of Hyperbolic Differential Equations 7(3) (2010) 383-404.

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  • Jean-Luc Guermond and Richard Pasquetti
    Entropy viscosity method for high-order approximations of conservation laws
    in "Spectral and High Order Methods for Partial Differential Equations (ICOSAHOM)", Lecture Notes in computational Science and Engineering, Vol. 76 (Jan S. Hesthaven and Einar M. Ronquist, eds.), Springer, Trondheim, Norway, June 2009 (2011) 411-418..

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  • U. Fjordholm, S. Mishra and E. Tadmor
    Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography
    J. Computational Physics 230(14) (2011) 5587-5609.

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  • Jean-Luc Guermond, Richard Pasquetti and Bojan Popov
    Entropy viscosity method for nonlinear conservation laws
    Journal of Computational Physics 230(11) (2011) 4248-4267.

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  • Ulrik S. Fjordholm
    Energy conservative and stable schemes for the two-layer shallow water equations
    in "Hyperbolic problems: Theory, Numerics and Applications", Vol. 2 (Tatsien Li and Song Jiang, eds.), Higher Education Press, Beijing, June 2010 (2011) 414-421..

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  • Gabriella Puppo and Matteo Semplice
    Numerical entropy and adaptivity for finite volume schemes
    Communications in Computational Physics 10(5) (2011) 1132-1160.

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  • U. Fjordholm and S. Mishra
    Accurate numerical discretizations of non-conservative hyperbolic systems
    Mathematical Modelling and Numerical Analysis 46(1) (2011) 187-206.

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  • Gabriella Puppo and Matteo Semplice
    Entropy and the numerical integration of conservation laws
    Physics Procedia (2011) 1-28.

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  • H. Kumar and S. Mishra
    Enropy stable numerical schemes for two-fluid plasma equations
    Journal of Scientific Computing 52 (2011) 401-425.

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  • Xiangxiong Zhang and Chi-Wang Shu
    A minimum entropy principle of high order schemes for gas dynamics equations
    Numerische Mathematik 121 (2011) 545-563.

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  • U. Fjordholm, S. Mishra and E. Tadmor
    Arbitrarily high order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws
    SIAM Journal on Numerical Analysis 50(2) (2012) 544-573.

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  • M. Svard and S. Mishra
    Entropy stable schemes for initial-boundary-value conservation laws
    Zeitschrift fĂĽr angewandte Mathematik und Physik (ZAMP) (2012).

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  • A. Mardane, U. Fjordholm, S. Mishra and E. Tadmor
    Entropy conservative and entropy stable finite volume schemes for multi-dimensional conservation laws on unstructured meshes,
    in "European Congress Computational Methods Applied Sciences and Engineering", Prceedings of ECCOMAS 2012 (J. Eberhardsteiner et. al. eds.), held in Vienna (2012).

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  • A. N. Mohammed and F. Ismail
    Study of an entropy-consistent Navier–Stokes flux
    International Journal of Computational Fluid Dynamics 27(1) (2012) 1-14.

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  • Jean-Luc Guermond and Bojan Popov
    Viscous regularization of the Euler equations and entropy principles.

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  • Manuel J. Castro, Ulrik S. Fjordholm, Siddhartha Mishra and Carlos ParĂ©s
    Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems
    SIAM J. Numerical Analysis 51(3) (2012) 1371-1391.

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  • A. Hiltebrand and S. Mishra
    Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws
    Numerische Mathematik 126(1) (2013) 103-151.

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  • Mark H. Carpenter and Travis C. Fisher
    High-order entropy stable formulations for computational fluid dynamics
    in "21st AIAA Computational Fluid Dynamics Conference", San-Diego, CA (2013) AIAA 2013-2868..

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  • M. Lukacova-Medvidova and N. Sfakianakis
    Entropy dissipation of moving mesh adaptation
    J. Hyperbolic Differential Equations (2014).

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  • Jean-Luc Guermond and Bojan Popov
    Entropy viscosity for the Euler equations and questions regarding parabolic regularization .
    in "Hyperbolic Problems: Hyperbolic Problems: Theory, Numerics, Applications", held in Padova in 2012 (Fabio Ancona, Alberto Bressan, Pierangelo Marcati, Andrea Marson eds.), AIMS on Applied Mathematics Vol. 8(1) (2014) 119-124..

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  • Philippe LeFloch and Sid Mishra
    Numerical methods with controlled dissipation for small-scale dependent shocks
    Acta Numerica 23 (2014) 743-816.

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  • M. Parsani, M. H. Carpenter and E. J. Nielsen
    Entropy stable wall boundary conditions for the compressible Navier-Stokes equations
    NASA Lengley Technical report TM–2014-218282 (2014).

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  • A. N. Gorban and D. J. Packwood
    Enhancement of the stability of lattice Boltzmann methods by dissipation control
    Physica A 414 (2014) 285-299.

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  • A. N. Mohammed and F. Ismail
    Entropy consistent methods for the Navier–Stokes equations. A first-order systems approach
    J. Scientific Computation 63 (2014) 612-631.

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  • M. Zakerzadeh and G. May
    On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws.

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  • Siddhartha Mishra and Laura V. Spinolo
    Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws
    Journal of Hyperbolic Differential Equations 12 (2015) 61-86.

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  • U. Fjordholm, R. Kappeli, S. Mishra and E. Tadmor
    Construction of approximate entropy measure valued solutions for systems of conservation laws
    Foundations of Computational Mathematics (2015).

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  • S. Lanthaler and S. Mishra
    Computation of measure valued solutions for the incompressible Euler equations
    Mathematical Models and Methods in Applied Sciences (2015).

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  • A. R. Winters and G. J. Gassner
    Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations
    Journal of Computational Physics 304 (2015) 72-108.

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  • E Tadmor
    Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws
    Discrete and Continuous Dynamical Systems-A 36(8) (2016) 4579-4598.

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  • D. Ray, P. Chandrashekar, U. S. Fjordholm and S. Mishra
    Entropy stable scheme on two-dimensional unstructured grids for Euler equations
    Communications in Computational Physics 19(5) (2016) 1111-1140.

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  • U. S. Fjordholm, S. Mishra and E. Tadmor
    On the computation of measure-valued solutions
    Acta Numerica 25 (2016) 567-679.

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  • Sudi Mungkasi
    Adaptive Finite Volume Method for the Shallow Water Equations on Triangular Grids
    Advances in Mathematical Physics 2016 (2016).

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  • Birte Schmidtmanna and Andrew R. Wintersb
    Hybrid entropy stable HLL-Type Riemann solvers for hyperbolic conservation laws.

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  • E. Tadmor
    Entropy stable schemes
    Handbook of Numerical Methods for Hyperbolic problems XVII (R. Abgrall and C.-W. Shu, eds.), Elsevier (2016) 467-493.

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