?. Hr, ?r , er , pr) 16: 17: u ? average(? l , ?r

E. Density, magnitude of B, magnitude of the velocity field, gas pressure and total pressure in the plane z = 0 obtained with the CT approach of Section 3.3 on a 257 3 grid, p.95

R. Abgrall and S. Karni, A Relaxation Scheme for the Two-Layer Shallow Water System, Hyperbolic Problems: Theory, pp.135-144
DOI : 10.1007/978-3-540-75712-2_11
URL : https://hal.archives-ouvertes.fr/inria-00334007

D. Aregba-driollet and C. Berthon, Numerical approximation of Kerr-Debye equations, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00293728

G. Arfken, Mathematical Methods for Physicists, 1985.
DOI : 10.1063/1.3034326

S. A. Balbus and J. C. Papaloizou, On the Dynamical Foundations of ?? Disks, The Astrophysical Journal, vol.521, issue.2, pp.650-658, 1999.
DOI : 10.1086/307594

D. S. Balsara, Linearized Formulation of the Riemann Problem for Adiabatic and Isothermal Magnetohydrodynamics, The Astrophysical Journal Supplement Series, vol.116, issue.1, p.119, 1998.
DOI : 10.1086/313092

D. S. Balsara, Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows, Journal of Computational Physics, vol.229, issue.6, pp.1970-1993, 2010.
DOI : 10.1016/j.jcp.2009.11.018
URL : http://arxiv.org/abs/0911.1613

D. S. Balsara, A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows, Journal of Computational Physics, vol.231, issue.22, pp.7476-7503, 2012.
DOI : 10.1016/j.jcp.2011.12.025

D. S. Balsara, M. Dumbser, and R. Abgrall, Multidimensional HLLC Riemann solver for unstructured meshes ??? With application to Euler and MHD flows, Journal of Computational Physics, vol.261, issue.1492, pp.172-208270, 1999.
DOI : 10.1016/j.jcp.2013.12.029

T. Barth, On the Role of Involutions in the Discontinuous Galerkin Discretization of Maxwell and Magnetohydrodynamic Systems, Compatible Spatial Discretizations The IMA Volumes in Mathematics and its Applications, pp.69-88, 2006.
DOI : 10.1007/0-387-38034-5_4

M. Baudin, C. Berthon, F. Coquel, R. Masson, and Q. H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Numerische Mathematik, vol.48, issue.3, pp.411-440, 2005.
DOI : 10.1007/s00211-004-0558-1

C. Berthon, Stability of the MUSCL Schemes for the Euler Equations, Communications in Mathematical Sciences, vol.3, issue.2, pp.133-157, 2005.
DOI : 10.4310/CMS.2005.v3.n2.a3

C. Berthon, Numerical approximations of the 10-moment Gaussian closure, Mathematics of Computation, vol.75, issue.256, pp.1809-1831, 2006.
DOI : 10.1090/S0025-5718-06-01860-6

C. Berthon, B. Braconnier, and B. Nkonga, Numerical approximation of a degenerated non-conservative multifluid model: relaxation scheme, International Journal for Numerical Methods in Fluids, vol.48, issue.1, pp.85-90, 2005.
DOI : 10.1002/fld.933

C. Berthon, M. Breuß, and M. Titeux, A relaxation scheme for the approximation of the pressureless Euler equations, Numerical Methods for Partial Differential Equations, vol.118, issue.2, pp.484-505, 2006.
DOI : 10.1002/num.20108

C. Berthon, P. Charrier, and B. Dubroca, An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions, Journal of Scientific Computing, issue.3, pp.31347-389, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00293559

C. Berthon and F. Marche, A Positive Preserving High Order VFRoe Scheme for Shallow Water Equations: A Class of Relaxation Schemes, SIAM Journal on Scientific Computing, vol.30, issue.5, pp.2587-2612, 2008.
DOI : 10.1137/070686147
URL : https://hal.archives-ouvertes.fr/hal-00370486

S. J. Billett and E. F. Toro, Unsplit WAF-Type Schemes for Three Dimensional Hyperbolic Conservation Laws, Numerical Methods for Wave Propagation of Fluid Mechanics and Its Applications, pp.75-124, 1998.
DOI : 10.1007/978-94-015-9137-9_4

A. Bonnement, T. Fajraoui, H. Guillard, M. Martin, A. Mouton et al., Finite volume method in curvilinear coordinates for hyperbolic conservation laws, ESAIM: Proc., volume, pp.163-176, 2011.
DOI : 10.1051/proc/2011019
URL : https://hal.archives-ouvertes.fr/hal-00914822

F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics. Birkhäuser, 2004.

J. U. Brackbill and D. C. Barnes, The Effect of Nonzero ??? ?? B on the numerical solution of the magnetohydrodynamic equations, Journal of Computational Physics, vol.35, issue.3, pp.426-430, 1980.
DOI : 10.1016/0021-9991(80)90079-0

M. Brio and C. C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, Journal of Computational Physics, vol.75, issue.2, pp.400-422, 1988.
DOI : 10.1016/0021-9991(88)90120-9

M. Brio, A. R. Zakharian, and G. M. Webb, Two-Dimensional Riemann Solver for Euler Equations of Gas Dynamics, Journal of Computational Physics, vol.167, issue.1, pp.177-195, 2001.
DOI : 10.1006/jcph.2000.6666

P. Cargo and G. Gallice, Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws, Journal of Computational Physics, vol.136, issue.2, pp.446-466, 1997.
DOI : 10.1006/jcph.1997.5773

C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, Numerische Mathematik, vol.36, issue.3, pp.451-478, 2005.
DOI : 10.1007/s00211-005-0612-7
URL : https://hal.archives-ouvertes.fr/hal-00112166

C. Chalons and J. Coulombel, Relaxation approximation of the Euler equations, Journal of Mathematical Analysis and Applications, vol.348, issue.2, pp.872-893, 2008.
DOI : 10.1016/j.jmaa.2008.07.034

T. Chang, G. Chen, and S. Yang, On the 2-D Riemann problem for the compressible Euler equations. I. Interaction of shocks and rarefaction waves. Discrete and Continuous Dynamical Systems, pp.555-584, 1995.

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion. Number v. 1 in Introduction to Plasma Physics and Controlled Fusion, 1984.
DOI : 10.1007/978-1-4757-5595-4

G. Chen, C. D. Levermore, and T. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Communications on Pure and Applied Mathematics, vol.44, issue.6, pp.47787-830, 1994.
DOI : 10.1002/cpa.3160470602
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.312

J. Chièze, Elements of Hydrodynamics Applied to the Interstellar Medium, In Starbursts Triggers, Nature, and Evolution, vol.9, pp.77-100, 1998.
DOI : 10.1007/978-3-662-29742-1_3

C. Chiosi, Hertzsprung?Russell Diagram, Encyclopedia of Astronomy and Astrophysics, 2000.
DOI : 10.1888/0333750888/1847

A. J. Chorin, Random choice solution of hyperbolic systems, Journal of Computational Physics, vol.22, issue.4, pp.517-533, 1976.
DOI : 10.1016/0021-9991(76)90047-4

F. Coquel and B. Perthame, Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics, SIAM Journal on Numerical Analysis, vol.35, issue.6, pp.2223-2249, 1998.
DOI : 10.1137/S0036142997318528

R. Courant, K. Friedrichs, R. Courant, K. Friedrichs, and H. Lewy, ???ber die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, vol.98, issue.6, pp.32-74, 1928.
DOI : 10.1007/BF01448839

P. Crispel, P. Degond, and M. Vignal, An asymptotically stable discretization for the Euler???Poisson system in the quasi-neutral limit, Comptes Rendus Mathematique, vol.341, issue.5, pp.323-328, 2005.
DOI : 10.1016/j.crma.2005.07.008

P. Crispel, P. Degond, and M. Vignal, An asymptotic preserving scheme for the two-fluid Euler???Poisson model in the quasineutral limit, Journal of Computational Physics, vol.223, issue.1, pp.208-234, 2007.
DOI : 10.1016/j.jcp.2006.09.004
URL : https://hal.archives-ouvertes.fr/hal-00635602

W. Dai and P. R. Woodward, On the Divergence???free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows, The Astrophysical Journal, vol.494, issue.1, p.317, 1998.
DOI : 10.1086/305176

P. A. Davidson, 11R45. Introduction to Magnetohydrodynamics. Cambridge Text in Applied Mathematics, Applied Mechanics Reviews, vol.55, issue.6, 2001.
DOI : 10.1115/1.1508153

H. Deconinck and R. Abgrall, Introduction to residual distribution methods. Lecture Series-Von Karman Institute for Fluid Dynamics, 2006.

H. Deconinck, K. Sermeus, and R. Abgrall, Status of multidimensional upwind residual distribution schemes and applications in aeronautics, Fluids 2000 Conference and Exhibit, 2000.
DOI : 10.2514/6.2000-2328

A. Dedner, F. Kemm, D. Kröner, C. Munz, T. Schnitzer et al., Hyperbolic Divergence Cleaning for the MHD Equations, Journal of Computational Physics, vol.175, issue.2, pp.645-673, 2002.
DOI : 10.1006/jcph.2001.6961
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.458.630

P. Degond, F. Deluzet, A. Sangam, and M. Vignal, An Asymptotic Preserving scheme for the Euler equations in a strong magnetic field, Journal of Computational Physics, vol.228, issue.10, pp.3540-3558, 2009.
DOI : 10.1016/j.jcp.2008.12.040
URL : https://hal.archives-ouvertes.fr/hal-00319630

P. Degond, S. Jin, and J. Liu, Mach-number uniform asymptotic-preserving gauge schemes for compressible flows, Bulletin of the Institute of Mathematics, vol.2, issue.4, pp.851-892, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00635618

P. Degond, J. Liu, and M. Vignal, Analysis of an Asymptotic Preserving Scheme for the Euler???Poisson System in the Quasineutral Limit, SIAM Journal on Numerical Analysis, vol.46, issue.3, pp.1298-1322, 2008.
DOI : 10.1137/070690584
URL : https://hal.archives-ouvertes.fr/hal-00635071

B. Després and C. Mazeran, Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems, Archive for Rational Mechanics and Analysis, vol.180, issue.3, pp.327-372, 2005.
DOI : 10.1007/s00205-005-0375-4

B. Einfeldt, On Godunov-Type Methods for Gas Dynamics, SIAM Journal on Numerical Analysis, vol.25, issue.2, pp.294-318, 1988.
DOI : 10.1137/0725021

B. Einfeldt, P. L. Roe, C. D. Munz, and B. Sjogreen, On Godunov-type methods near low densities, Journal of Computational Physics, vol.92, issue.2, pp.273-295, 1991.
DOI : 10.1016/0021-9991(91)90211-3

C. R. Evans and J. F. Hawley, Simulation of magnetohydrodynamic flows - A constrained transport method, The Astrophysical Journal, vol.332, issue.332, pp.659-677, 1988.
DOI : 10.1086/166684

L. C. Evans, Partial Differential Equations. Graduate studies in mathematics, 2010.

R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, Handbook of Numerical Analysis, vol.7, pp.713-1018, 2000.
URL : https://hal.archives-ouvertes.fr/hal-00346077

S. Fabre, Stability analysis of the Euler-poisson equations, Journal of Computational Physics, vol.101, issue.2, pp.445-451, 1992.
DOI : 10.1016/0021-9991(92)90020-Y

M. Fey and M. Torrilhon, A constrained transport upwind scheme for divergencefree advection, Hyperbolic Problems: Theory, Numerics, Applications, pp.529-538, 2003.

A. Frank, T. W. Jones, D. Ryu, and J. B. Gaalaas, The Magnetohydrodynamic Kelvin-Helmholtz Instability: A Two-dimensional Numerical Study, The Astrophysical Journal, vol.460, p.777, 1996.
DOI : 10.1086/177009

S. Fromang, P. Hennebelle, and R. Teyssier, A high order Godunov scheme with constrained transport and??adaptive mesh refinement for astrophysical magnetohydrodynamics, Astronomy and Astrophysics, vol.457, issue.2, pp.371-384, 2006.
DOI : 10.1051/0004-6361:20065371

J. M. Gallardo, C. Parés, and M. Castro, On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, Journal of Computational Physics, vol.227, issue.1, pp.574-601, 2007.
DOI : 10.1016/j.jcp.2007.08.007

T. Gallouët, J. Hérard, and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Computers & Fluids, vol.32, issue.4, pp.479-513, 2003.
DOI : 10.1016/S0045-7930(02)00011-7

T. A. Gardiner and J. M. Stone, An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, Journal of Computational Physics, vol.227, issue.8, pp.4123-4141, 2008.
DOI : 10.1016/j.jcp.2007.12.017

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Communications on Pure and Applied Mathematics, vol.12, issue.4, pp.697-715, 1965.
DOI : 10.1002/cpa.3160180408

J. Glimm and P. D. Lax, Decay of solutions of systems of hyperbolic conservation laws, Bulletin of the American Mathematical Society, vol.73, issue.1, pp.105-105, 1967.
DOI : 10.1090/S0002-9904-1967-11666-5

E. Godlewski and P. Raviart, Numerical approximation of hyperbolic systems of conservation laws, 1996.
DOI : 10.1007/978-1-4612-0713-9

S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, pp.271-306, 1959.

M. González, E. Audit, and P. Huynh, HERACLES: a three-dimensional radiation hydrodynamics code, Astronomy and Astrophysics, vol.464, issue.2, pp.429-435, 2007.
DOI : 10.1051/0004-6361:20065486

J. J. Gottlieb and C. P. Groth, Assessment of riemann solvers for unsteady one-dimensional inviscid flows of perfect gases, Journal of Computational Physics, vol.78, issue.2, pp.437-458, 1988.
DOI : 10.1016/0021-9991(88)90059-9

J. Greenberg, A. Leroux, R. Baraille, and A. Noussair, Analysis and Approximation of Conservation Laws with Source Terms, SIAM Journal on Numerical Analysis, vol.34, issue.5, pp.1980-2007, 1997.
DOI : 10.1137/S0036142995286751

J. M. Greenberg and A. Y. Leroux, A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations, SIAM Journal on Numerical Analysis, vol.33, issue.1, pp.1-16, 1996.
DOI : 10.1137/0733001

P. M. Gresho and S. T. Chan, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: Implementation, International Journal for Numerical Methods in Fluids, vol.7, issue.5, pp.621-659, 1990.
DOI : 10.1002/fld.1650110510

H. Guillard and E. Daniel, A well balanced scheme for gas flows in protoplanetary nebulae, Finite Volumes for Complex Applications IV, pp.355-364, 2005.

K. Gurski, An HLLC-Type Approximate Riemann Solver for Ideal Magnetohydrodynamics, SIAM Journal on Scientific Computing, vol.25, issue.6, pp.2165-2187, 2004.
DOI : 10.1137/S1064827502407962

A. Harten and P. Lax, A Random Choice Finite Difference Scheme for Hyperbolic Conservation Laws, SIAM Journal on Numerical Analysis, vol.18, issue.2, pp.289-315, 1981.
DOI : 10.1137/0718021

A. Harten, P. Lax, and B. Van-leer, On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws, SIAM Review, vol.25, issue.1, pp.35-61, 1983.
DOI : 10.1137/1025002

C. Helzel, J. A. Rossmanith, and B. Taetz, An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations, Journal of Computational Physics, vol.230, issue.10, pp.3803-3829, 2011.
DOI : 10.1016/j.jcp.2011.02.009

T. Y. Hou and P. G. Lefloch, Why nonconservative schemes converge to wrong solutions: error analysis, Mathematics of Computation, vol.62, issue.206, pp.497-530, 1994.
DOI : 10.1090/S0025-5718-1994-1201068-0

T. J. Hughes, L. Mazzei, and K. E. Jansen, Large Eddy Simulation and the variational multiscale method, Computing and Visualization in Science, vol.3, issue.1-2, pp.47-59, 2000.
DOI : 10.1007/s007910050051

S. Inaba, P. Barge, E. Daniel, and H. Guillard, A two-phase code for protoplanetary disks, Astronomy and Astrophysics, vol.431, issue.1, pp.365-379, 2005.
DOI : 10.1051/0004-6361:20041085
URL : https://hal.archives-ouvertes.fr/hal-00017327

R. A. James, The solution of poisson's equation for isolated source distributions, Journal of Computational Physics, vol.25, issue.2, pp.71-93, 1977.
DOI : 10.1016/0021-9991(77)90013-4

P. Janhunen, A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods, Journal of Computational Physics, vol.160, issue.2, pp.649-661, 2000.
DOI : 10.1006/jcph.2000.6479

G. Jiang and C. Shu, Efficient Implementation of Weighted ENO Schemes, Journal of Computational Physics, vol.126, issue.1, pp.202-228, 1996.
DOI : 10.1006/jcph.1996.0130

S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, ESAIM: Mathematical Modelling and Numerical Analysis, vol.35, issue.4, pp.631-645, 2001.
DOI : 10.1051/m2an:2001130

S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics, vol.54, issue.3, pp.235-276, 1995.
DOI : 10.1002/cpa.3160480303

R. Kippenhahn and A. Weigert, Stellar structure and evolution. Astronomy and Astrophysics Library, 1990.

A. Kurganov, S. Noelle, and G. Petrova, Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations, SIAM Journal on Scientific Computing, vol.23, issue.3, pp.707-740, 2000.
DOI : 10.1137/S1064827500373413

A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numerical Methods for Partial Differential Equations, vol.160, issue.5, pp.584-608, 2002.
DOI : 10.1002/num.10025

P. Lax and X. Liu, Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes, SIAM Journal on Scientific Computing, vol.19, issue.2, pp.319-340, 1998.
DOI : 10.1137/S1064827595291819

P. Lax and B. Wendroff, Systems of conservation laws, Communications on Pure and Applied Mathematics, vol.47, issue.2, pp.217-237, 1960.
DOI : 10.1002/cpa.3160130205

R. J. Leveque, Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm, Journal of Computational Physics, vol.146, issue.1, pp.346-365, 1998.
DOI : 10.1006/jcph.1998.6058

R. J. Leveque, Finite volume methods for hyperbolic problems, 2002.
DOI : 10.1017/CBO9780511791253

R. J. Leveque and D. S. Bale, Wave Propagation Methods for Conservation Laws with Source Terms, Proceedings of the 7th International Conference on Hyperbolic Problems, pp.609-618, 1998.
DOI : 10.1007/978-3-0348-8724-3_12

R. J. Leveque and M. Pelanti, A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes, Journal of Computational Physics, vol.172, issue.2, pp.572-591, 2001.
DOI : 10.1006/jcph.2001.6838
URL : https://hal.archives-ouvertes.fr/hal-01342280

H. Li, S. A. Colgate, B. Wendroff, and R. Liska, Rossby Wave Instability of Thin Accretion Disks. III. Nonlinear Simulations, The Astrophysical Journal, vol.551, issue.2, p.874, 2001.
DOI : 10.1086/320241

H. Li, J. M. Finn, R. V. Lovelace, and S. A. Colgate, Rossby Wave Instability of Thin Accretion Disks. II. Detailed Linear Theory, The Astrophysical Journal, vol.533, issue.2, p.1023, 2000.
DOI : 10.1086/308693

J. Li, Z. Yang, and Y. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, Journal of Differential Equations, vol.250, issue.2, pp.782-798, 2011.
DOI : 10.1016/j.jde.2010.07.009

Y. Li, Convergence of the nonisentropic Euler???Poisson equations to incompressible type Euler equations, Journal of Mathematical Analysis and Applications, vol.342, issue.2, pp.1107-1125, 2008.
DOI : 10.1016/j.jmaa.2007.12.067

T. Linde, A practical, general-purpose, two-state HLL Riemann solver for hyperbolic conservation laws, International Journal for Numerical Methods in Fluids, vol.277, issue.3-4, pp.3-4391, 2002.
DOI : 10.1002/fld.312

R. Liska and B. Wendroff, Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations, SIAM Journal on Scientific Computing, vol.25, issue.3, pp.995-1017, 2003.
DOI : 10.1137/S1064827502402120

T. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Communications on Pure and Applied Mathematics, vol.2, issue.12, pp.1553-1586, 1999.
DOI : 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S

P. Londrillo and L. D. Zanna, High???Order Upwind Schemes for Multidimensional Magnetohydrodynamics, The Astrophysical Journal, vol.530, issue.1, p.508, 2000.
DOI : 10.1086/308344
URL : http://arxiv.org/abs/astro-ph/9910086

P. Londrillo and L. D. Zanna, On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, Journal of Computational Physics, vol.195, issue.1, pp.17-48, 2004.
DOI : 10.1016/j.jcp.2003.09.016

J. Luo, K. Xu, and N. Liu, A Well-Balanced Symplecticity-Preserving Gas-Kinetic Scheme for Hydrodynamic Equations under Gravitational Field, SIAM Journal on Scientific Computing, vol.33, issue.5, pp.2356-2381, 2011.
DOI : 10.1137/100803699

P. Maire, R. Abgrall, J. Breil, and J. Ovadia, A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems, SIAM Journal on Scientific Computing, vol.29, issue.4, pp.1781-1824, 2007.
DOI : 10.1137/050633019
URL : https://hal.archives-ouvertes.fr/inria-00334022

M. and B. Nkonga, Multi-scale Godunov-type method for cellcentered discrete Lagrangian hydrodynamics, Journal of Computational Physics, vol.228, issue.3, pp.799-821, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00290717

A. Malagoli, G. Bodo, and R. Rosner, On the Nonlinear Evolution of Magnetohydrodynamic Kelvin-Helmholtz Instabilities, The Astrophysical Journal, vol.456, p.708, 1996.
DOI : 10.1086/176691

A. Mignone, P. Tzeferacos, and G. Bodo, High-order conservative finite difference GLM???MHD schemes for cell-centered MHD, Journal of Computational Physics, vol.229, issue.17, pp.5896-5920, 2010.
DOI : 10.1016/j.jcp.2010.04.013

T. Miyoshi and K. Kusano, A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, Journal of Computational Physics, vol.208, issue.1, pp.315-344, 2005.
DOI : 10.1016/j.jcp.2005.02.017

C. Munz, P. Omnes, R. Schneider, E. Sonnendrücker, and U. Voß, Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model, Journal of Computational Physics, vol.161, issue.2, pp.484-511, 2000.
DOI : 10.1006/jcph.2000.6507

C. Munz, R. Schneider, E. Sonnendrücker, and U. Voss, Maxwell's equations when the charge conservation is not satisfied, Comptes Rendus de l'Académie des Sciences -Series I -Mathematics, pp.431-436, 1999.
DOI : 10.1016/S0764-4442(99)80185-2

R. Natalini and F. Rousset, Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations, Proceedings of the American Mathematical Society, pp.2251-2258, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00458157

S. Noelle, N. Pankratz, G. Puppo, and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, Journal of Computational Physics, vol.213, issue.2, pp.474-499, 2006.
DOI : 10.1016/j.jcp.2005.08.019

K. G. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), 1994.

D. J. Price and J. J. Monaghan, Smoothed Particle Magnetohydrodynamics -- III. Multidimensional tests and the ??{middle dot}B= 0 constraint, Monthly Notices of the Royal Astronomical Society, vol.364, issue.2, pp.384-406, 2005.
DOI : 10.1111/j.1365-2966.2005.09576.x

J. J. Quirk, A contribution to the great Riemann solver debate, International Journal for Numerical Methods in Fluids, vol.1, issue.6, pp.555-574, 1994.
DOI : 10.1002/fld.1650180603

J. J. Quirk, Godunov-Type Schemes Applied to Detonation Flows, Combustion in High-Speed Flows, pp.575-596, 1994.
DOI : 10.1007/978-94-011-1050-1_21

R. D. Richtmyer and K. W. Morton, Difference methods for initial-value problems. Number 4, 1967.

P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, vol.43, issue.2, pp.357-372, 1981.
DOI : 10.1016/0021-9991(81)90128-5

P. L. Roe, Characteristic-Based Schemes for the Euler Equations, Annual Review of Fluid Mechanics, vol.18, issue.1, pp.337-365, 1986.
DOI : 10.1146/annurev.fl.18.010186.002005

J. Rossmanith, An Unstaggered, High???Resolution Constrained Transport Method for Magnetohydrodynamic Flows, SIAM Journal on Scientific Computing, vol.28, issue.5, pp.1766-1797, 2006.
DOI : 10.1137/050627022

R. Saurel, M. Larini, and J. C. Loraud, Exact and Approximate Riemann Solvers for Real Gases, Journal of Computational Physics, vol.112, issue.1, pp.126-137, 1994.
DOI : 10.1006/jcph.1994.1086

M. Schleicher, Eine einfaches und effizientes verfahren zur lösung des Riemannproblems, Zeitschrift für Flugwissenschaften und Weltraumforschung, pp.265-269, 1993.

C. W. Schulz-rinne, J. P. Collins, and H. M. Glaz, Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics, SIAM Journal on Scientific Computing, vol.14, issue.6, pp.1394-1414, 1993.
DOI : 10.1137/0914082

L. I. Sedov, Propagation of strong blast waves, Prikl. Mat. Mekh, vol.10, issue.2, pp.241-250, 1946.

J. M. Stone and T. Gardiner, A simple unsplit Godunov method for multidimensional MHD, New Astronomy, vol.14, issue.2, pp.139-148, 2009.
DOI : 10.1016/j.newast.2008.06.003

I. Suliciu, On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure, International Journal of Engineering Science, vol.28, issue.8, pp.829-841, 1990.
DOI : 10.1016/0020-7225(90)90028-H

I. Suliciu, Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations, International Journal of Engineering Science, vol.30, issue.4, pp.483-494, 1992.
DOI : 10.1016/0020-7225(92)90039-J

A. Suresh, Positivity-Preserving Schemes in Multidimensions, SIAM Journal on Scientific Computing, vol.22, issue.4, pp.1184-1198, 2000.
DOI : 10.1137/S1064827599360443

R. Teyssier, S. Fromang, and E. Dormy, Kinematic dynamos using constrained transport with high order Godunov schemes and adaptive mesh refinement, Journal of Computational Physics, vol.218, issue.1, pp.44-67, 2006.
DOI : 10.1016/j.jcp.2006.01.042
URL : http://arxiv.org/abs/astro-ph/0601715

E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: A practical introduction, 2009.

G. Tóth, The ?????B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes, Journal of Computational Physics, vol.161, issue.2, pp.605-652, 2000.
DOI : 10.1006/jcph.2000.6519

P. Váchal, R. Liska, and B. Wendroff, Fully Two-dimensional HLLEC Riemann Solver and Associated Difference Schemes, Numerical Mathematics and Advanced Applications ENUMATH 2003, pp.815-824, 2004.
DOI : 10.1007/978-3-642-18775-9_80

S. Van-criekingen, E. Audit, J. Vides, and B. Braconnier, Time-implicit hydrodyanmics for Euler flows, ESAIM Proc. of the SMAI 2013 Congress, 2014.

B. Van-leer, Towards the ultimate conservative difference scheme I. The quest of monotonicity, Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, pp.163-168, 1973.
DOI : 10.1007/BFb0118673

B. Van-leer, Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme, Journal of Computational Physics, vol.14, issue.4, pp.361-370, 1974.
DOI : 10.1016/0021-9991(74)90019-9

B. Van-leer, Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow, Journal of Computational Physics, vol.23, issue.3, pp.263-275, 1977.
DOI : 10.1016/0021-9991(77)90094-8

B. Van-leer, Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection, Journal of Computational Physics, vol.23, issue.3, pp.276-299, 1977.
DOI : 10.1016/0021-9991(77)90095-X

B. Van-leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, Journal of Computational Physics, vol.32, issue.1, pp.101-136, 1979.
DOI : 10.1016/0021-9991(79)90145-1

J. Vides, E. Audit, H. Guillard, and B. Nkonga, Divergence-free MHD Simulations with the HERACLES Code, ESAIM: Proc., volume, pp.180-194, 2013.
DOI : 10.1051/proc/201343012
URL : https://hal.archives-ouvertes.fr/hal-01061303

J. Vides, E. Audit, and B. Nkonga, A relaxation scheme for inviscid flows under gravitational influence, ESAIM Proc. of the SMAI 2013 Congress, 2014.
DOI : 10.1051/proc/201445054
URL : https://hal.archives-ouvertes.fr/hal-01103515

J. Vides, B. Braconnier, E. Audit, C. Berthon, and B. Nkonga, Abstract, Communications in Computational Physics, vol.94, issue.01, pp.46-75, 2014.
DOI : 10.1086/307594

J. Vides, B. Nkonga, and E. Audit, A simple two-dimensional extension of the HLL Riemann solver for gas dynamics, INRIA Research Report, issue.8540, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00998235

J. Vides, B. Nkonga, E. Audit, S. Van-criekingen, M. Szydlarski et al., A simple two-dimensional extension of the HLL Riemann solver for hyperbolic systems of conservation laws A Godunov-type solver for gravitational flows: Towards a time-implicit version in the HERACLES code, 8th International Conference of Numerical Modeling of Space Plasma Flows, pp.279-284, 2013.

J. Vonneumann and R. D. Richtmyer, A Method for the Numerical Calculation of Hydrodynamic Shocks, Journal of Applied Physics, vol.21, issue.3, pp.232-237, 1950.
DOI : 10.1063/1.1699639

B. Wendroff, A two-dimensional HLLE Riemann solver and associated Godunovtype difference scheme for gas dynamics, Computers & Mathematics with Applications, vol.38, pp.11-12175, 1999.

G. B. Whitham, Linear and nonlinear waves, 1974.

P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics, vol.54, issue.1, pp.115-173, 1984.
DOI : 10.1016/0021-9991(84)90142-6

Y. Xing and C. Shu, High Order Well-Balanced WENO Scheme for the Gas Dynamics Equations Under Gravitational Fields, Journal of Scientific Computing, vol.143, issue.2-3, pp.645-662, 2013.
DOI : 10.1007/s10915-012-9585-8

K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas and Propagation, pp.302-307, 1966.

T. Zhang and Y. Zheng, Conjecture on the Structure of Solutions of the Riemann Problem for Two-Dimensional Gas Dynamics Systems, SIAM Journal on Mathematical Analysis, vol.21, issue.3, pp.593-630, 1990.
DOI : 10.1137/0521032

Y. Zheng, Systems of conservation laws: Two-dimensional Riemann problems, Progress in nonlinear differential equations and their applications. Birkhäuser, 2001.
DOI : 10.1007/978-1-4612-0141-0