[
[1] P. Arminjon, M.C. Viallon, A. Madrane and H. Kaddouri, Discontinuous finite elements and a 2-dimensional finite volume generalization of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows on unstructured grids, CFD Review, M. Hafez and K. Oshima, editors, John Wiley, pp. 241–261, (1997).10.1142/9789812812957_0012
]Search in Google Scholar
[
[2] T.J. Barth, Numerical methods for gaz-dynamics systems on unstructured meshes. In An introduction to Recent Developments in Theory and Numerics of Conservation Laws pp 195-285. Lecture Notes in Computational Science and Engineering Volume 5, Springer, Berlin. Eds D. Kroner, M. Ohlberger and Rohde, C., (1999).10.1007/978-3-642-58535-7_5
]Search in Google Scholar
[
[3] M. W. Bohm, A. R. Winters, G. J. Gassner, D. Derigs, F. J. Hindenlang and J. Saur. An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification. Journal of Computational Physics, to appear. DOI: 10.1016/j.jcp.2018.06.027.10.1016/j.jcp.2018.06.027
]Search in Google Scholar
[
[4] M. H. Carpenter, T. C. Fisher, E. J. Nielsen and S. H. Frankel. Entropy stable spectral collocation schemes for the NavierStokes equations: discontinuous interfaces. SIAM Journal on Scientific Computing, 36:B835B867, 2014.10.1137/130932193
]Search in Google Scholar
[
[5] T. Chen and C.-W. Shu. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. Journal of Computational Physics, 345:427461, 2017.10.1016/j.jcp.2017.05.025
]Search in Google Scholar
[
[6] G. Chavent and J. Jaffré and R. Eymard and D. Guérillot and L.Weill, Discontinuous and mixed finite elements for two-phase incompressible flow, SPE Reservoir Engineering, 5, 567–575, (1990).10.2118/16018-PA
]Search in Google Scholar
[
[7] G. Chavent and J. Jaffré, Mathematical Models and Finite Element for Reservoir Simulation ”, ”North Holland, Amsterdam, (1986).
]Search in Google Scholar
[
[8] B. Cockburn, S. Hou and C.W. Shu, The Runge-Kutta local projection discontinuous Galerkin Finite Element Method for conservation laws IV : the multi-dimensional case, Math. Comp. Vol. 54, No. 190. 545–581, (1990).10.1090/S0025-5718-1990-1010597-0
]Search in Google Scholar
[
[9] Cockburn, B., Lin, S-y, Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84, 90–113, (1989).10.1016/0021-9991(89)90183-6
]Search in Google Scholar
[
[10] P. Collela, Multi-dimensional upwind methods for hyperbolic conservation laws, J. Comput. Physics, 87, 171–200, (1990).10.1016/0021-9991(90)90233-Q
]Search in Google Scholar
[
[11] C. Dafrmos, Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, (2000).10.1007/978-3-662-22019-1
]Search in Google Scholar
[
[12] J.-A. Désidéri and A. Dervieux, Compressible flow solvers using unstructured grids. In Computational fluid dynamics, Vol. 1, 2, volume 88 of von Karman Inst. Fluid Dynam. Lecture Ser., page 115. von Karman Inst. Fluid Dynamics, Rhode-St- Gense, (1988).
]Search in Google Scholar
[
[13] U.S. Fjordholm, S. Mishra and E. Tadmor, Energy preserving and energy stable schemes for the shallow water equations. “Foundations of Computational Mathematics”, Proc. FoCM held in Hong Kong 2008 (F. Cuckers, A. Pinkus and M. Todd, eds), London Math. Soc. Lecture Notes Ser. 363, pp. 93–139, (2009).10.1017/CBO9781139107068.005
]Search in Google Scholar
[
[14] Fjordholm, U.S., Mishra, S., Tadmor, E., Arbitrary order accurate essentially non-oscillatory entropy stable schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573, (2012).10.1137/110836961
]Search in Google Scholar
[
[15] G. J. Gassner, A. R. Winters, F. J. Hindenlang, and D. A. Kopriva. The BR1 scheme is stable for the compressible NavierStokes equations. Journal of Scientific Computing, 77:154200, 2018.10.1007/s10915-018-0702-1
]Search in Google Scholar
[
[16] Harten, A., Engquist, B., Osher, S., Chakravarty, S.R., Uniformly high order accurate essentially nonoscillatory schemes, III. J. Comput. Phys. 71, 231–303, (1987).10.1016/0021-9991(87)90031-3
]Search in Google Scholar
[
[17] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, (1996).10.1007/978-1-4612-0713-9
]Search in Google Scholar
[
[18] G.D.V. Gowda, Discontinuous finite element for nonlinear scalar conservation laws, Thèse de Doctorat, Universit é Paris IX-Dauphine, (1988).
]Search in Google Scholar
[
[19] INRIA and GAMNI-SMAI, Workshop on hypersonic flows for reentry problems, Problem 6 : Flow over a double ellipse, test case 6.1 : Non-Reactive Flows. Antibes, France, January 22-25, (1990)
]Search in Google Scholar
[
[20] F. Ismail and P. L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys., vol. 228, issue 15, pp. 5410–5436, (2009).10.1016/j.jcp.2009.04.021
]Search in Google Scholar
[
[21] B.V. Leer, Towards the ultimate conservative scheme: IV. A new approach to numerical convection, J. Comput. Physics, 23, 276–299, (1977).10.1016/0021-9991(77)90095-X
]Search in Google Scholar
[
[22] P. Lesaint and P. Raviart, On a finite element method for solving the neutron transport equations, in Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, (1974).10.1016/B978-0-12-208350-1.50008-X
]Search in Google Scholar
[
[23] Y. Liu, C.-W. Shu and M. Zhang. Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. Journal of Computational Physics, 354:163178, 2018.10.1016/j.jcp.2017.10.043
]Search in Google Scholar
[
[24] A. Madrane, E.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 and A. Tzavaras, eds), AMS Proc. Symp. Applied Math., University of Marylan, 775–784, (2009).
]Search in Google Scholar
[
[25] A. Madrane, U. S. Fjordholm, S. Mishra, E. Tadmor, Entropy conservative and entropy stable finite volume schemes for multi-dimensional conservation laws on unstructured meshes, ECCOMAS 2012, Proc. J. Eberhardsteiner et al. (eds.), Vienna, Austria, September 10-14, (2012).
]Search in Google Scholar
[
[26] A. Madrane, S. Mishra, E. Tadmor, Entropy conservative and entropy stable finite volume/finite element schemes for the Navier-Stokes equations on unstructured meshes, ECCOMAS 2014, Proc. E. Onãte et al. (eds.), Barcelona Spain, Jul. 20, (2014).
]Search in Google Scholar
[
[27] S. Osher and F. Salomon, Upwind difference schemes for hyperbolic systems of conservation laws, Mathematics of Computation, 38, 158, 339–374, (1982).10.1090/S0025-5718-1982-0645656-0
]Search in Google Scholar
[
[28] J. Peraire and M. Vahdati and K. Morgan and O.C. Zienkiewicz, Adaptive remeshing for compressible flow computations, J. Comput. Physics, 72, 449–466, (1987).10.1016/0021-9991(87)90093-3
]Search in Google Scholar
[
[29] J. Peraire and L. Formaggia and J. Peiro and K. Morgan and O. C. Zienkiewicz, Finite element Euler computation in three dimensions, AIAA paper, 87-0032, (1987).
]Search in Google Scholar
[
[30] W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Tech.Rep. LA-UR-73-479, Los Alamos Scientific Laboratory, (1973).
]Search in Google Scholar
[
[31] F. Renac. Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows. Journal of Computational Physics, 382:126, 2019.10.1016/j.jcp.2018.12.035
]Search in Google Scholar
[
[32] P. L Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes, Journal of Computational Physics, 43, 357–372, (1981).10.1016/0021-9991(81)90128-5
]Search in Google Scholar
[
[33] Shu, C.W., Osher, S., Efficient implementation of essentially non-oscillatory schemesII. J. Comput. Phys. 83, 32–78, (1989).10.1016/0021-9991(89)90222-2
]Search in Google Scholar
[
[34] Z. Sun, J. A. Carillo and C.-W. Shu. A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials. Journal of Computational Physics, 352:76104, 2018.10.1016/j.jcp.2017.09.050
]Search in Google Scholar
[
[35] Z. Sun, J. A. Carillo and C.-W. Shu. An entropy stable high-order discontinuous Galerkin method for crossdiffusion gradient flow systems. Kinetic and Related Models, 12:885908, 2019.10.3934/krm.2019033
]Search in Google Scholar
[
[36] E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp., 49, 91–103, (1987).10.1090/S0025-5718-1987-0890255-3
]Search in Google Scholar
[
[37] E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time–dependant problem, Acta Numer., 12, 451–512, (2003).10.1017/S0962492902000156
]Search in Google Scholar
