. [. Bibliographie and . Abgrall, How to Prevent Pressure Oscillations in multicomponent Flow Calculations: A Quasi-conservative Approach, J.C.P, vol.125, pp.150-160, 1996.

[. Abgrall and R. Saurel, A Multiphase Godunov Method for Compressible Multiuid and Multiphase Flows, J.C.P, vol.150, pp.425-467, 1999.

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

N. Aslan, NUMERICAL SOLUTIONS OF ONE-DIMENSIONAL MHD EQUATIONS BY A FLUCTUATION APPROACH, International Journal for Numerical Methods in Fluids, vol.100, issue.7, pp.569-580, 1996.
DOI : 10.1002/(SICI)1097-0363(19960415)22:7<569::AID-FLD246>3.0.CO;2-7

[. Balsara, D. S. Spicer, and D. , Maintaining Pressure Positivity in Magnetohydrodynamic Simulations, Maintaining Pressure Positivity in Magnetohydrodynamic Simulations, pp.133-148, 1999.
DOI : 10.1006/jcph.1998.6108

[. Balsara, D. S. Spicer, and D. , A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations, Journal of Computational Physics, vol.149, issue.2, pp.270-292, 1999.
DOI : 10.1006/jcph.1998.6153

. Ch and . Berthon, A combustion model and approximate nonlinear projection scheme, 2001.

. Be-co-]-ch and F. Berthon, Nonlinear projection methods for multi-entropies Navier- Stokes systems, 2001.

. A. Ber-vaz-], M. E. Bermudez, and . Vazquez, Upwind Methods for Hyperbolic Conservation Laws with Source Terms, Computers and Fluids, vol.23, pp.1049-1071, 1994.

F. Bez-]-bezard, F. Bezard, and B. Despres, An Entropic Solver for Ideal Lagrangian Magnetohydrodynamics, Journal of Computational Physics, vol.154, issue.1, pp.65-89, 1998.
DOI : 10.1006/jcph.1999.6300

F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, Journal of Statistical Physics, vol.95, issue.1/2, pp.113-170, 1999.
DOI : 10.1023/A:1004525427365

S. Brassier, Résolution numérique d'un modèle MHD à deux températures et loi d'Ohm généralisée : solveur de Roe, suivi d'interface, équations de transport non linéaires à coecients discontinus. Modélisation d'un commutateur à plasma, 1998.

[. Brassier and G. Gallice, A Roe-scheme for the Bi-temperature Model of Magnetohydrodynamics, Computers and Mathematics with Applications, pp.41-257, 2001.

[. Brio, M. Wu, and C. C. , 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

P. Cg-]-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

F. Coquel, E. Godlewski, and B. Perthame, Some new Godunov and Relaxation Methods for two Phase Flows Problems, Godunov Methods: Theory and Applications Construction of a Roe linearization for the ideal MHD equations, C. R. Acad. Sci. Paris, vol.323, pp.951-955, 1996.

[. Cargo and A. Y. Leroux, Un schéma équilibre adapté au modèle d'atmosphère avec termes de gravité, C. R. Acad. Sci, pp.73-876, 1994.

. M. Cmp, J. Castro, C. Macias, and . Pares, A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1D shallow water system, Math. Model. and Numer. Anal, vol.35, pp.107-127, 2001.

[. F. Colombeau, A. Y. Leroux, A. Noussair, and B. Perrot, Microscopic Profiles of Shock Waves and Ambiguities in Multiplications of Distributions, SIAM Journal on Numerical Analysis, vol.26, issue.4, pp.871-883, 1989.
DOI : 10.1137/0726048

S. Cordier, Modélisation mathématique et simulation numérique du plasma magnétosphérique, Thèse de l'Ecole Normale Supérieure de Cachan, 1994.

J. P. Croisille, R. Khanr, and G. Chanteur, Numerical simulation of the MHD equations by a kinetic-type method, Journal of Scientific Computing, vol.99, issue.1, pp.481-492, 1996.
DOI : 10.1007/BF02087961

[. Coquel, F. Liou, and M. S. , Hybrid Upwind Splitting (HUS) by a eld by eld Decomposition, p.106843, 1995.

[. Coquel, F. Marmignon, and C. , A Roe-type linearizationfor the Euler equations for weakly ionized multi-component and multi-temperature gas, AIAA Paper, pp.95-1675, 1995.

W. Dai and P. Woodward, A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations, Journal of Computational Physics, vol.142, issue.2, pp.331-369, 1998.
DOI : 10.1006/jcph.1998.5944

. B. Da-pi-]-r, J. M. Dahlburg, and . Picone, Evolution of the Orszag-Tag vortex system in a compressible medium, Phys. of Fluids B, vol.1, p.2153, 1989.

P. [. Dal-maso, F. Floch, and . Murat, Denition and weak stability of a non conservative product, J. Maths Pures Appl, vol.74, pp.483-548, 1995.

P. Degond, P. F. Peyrard, G. Russo, and P. Villedieu, Polynomial upwind schemes for hyperbolic systems, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.328, issue.6, 1999.
DOI : 10.1016/S0764-4442(99)80194-3

B. Després, In??galit?? entropique pour un solveur conservatif du syst??me de la dynamique des gaz en coordonn??es de Lagrange, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.324, issue.11, pp.1301-1306, 1997.
DOI : 10.1016/S0764-4442(99)80417-0

B. Després, Lagrangian systems of conservation laws, Numerische Mathematik, vol.89, issue.1
DOI : 10.1007/PL00005465

. Dubois-]-b and . Dubois, Systems of Conservation Laws invariant for Galileo group and space reexion, CNAM, note 318, 1999.

[. Dubroca and G. Gallice, Problème mixte hyperbolique pour un système de lois de conservation monodimensionnel, C. R. Acad. Sciences Paris, pp.317-320, 1988.

[. Dubroca and G. Gallice, Résultats d'existence et d'unicité du problème mixte pour des systèmes hyperboliques de lois de conservation , Communication in Partial Dierential Equation, pp.59-80, 1988.

. P. Fmo-]-r, B. Fedwik, S. Merriman, and . Osher, Simplied Discretization of Systems of Hyperbolic Conservation Laws Containing Advection Equations, J.C.P, vol.157, pp.302-326, 2000.

. Flandrin-]-l and . Flandrin, Méthodes cell-centered pour l'approximation des équations de Navier-Stokes sur des maillages non structurés, 1995.

G. Gallice, Système d'Euler-Poisson, magnétohydrodynamique et schéma de Roe, Thèse de l, 1997.

. [. Gallice, Simulation of the Collision of Jupiter and the Shoemaker-Levi 9 Comet with a TVD Roe Type Scheme Preserving the Atmospheric Equilibrium in the Unperturbed Regions, 1995.

G. Gallice, Sch??mas de type Godunov entropiques et positifs pour la dynamique des gaz et la magn??tohydrodynamique lagrangiennes, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.332, issue.11, pp.1037-1040, 2001.
DOI : 10.1016/S0764-4442(01)01968-1

. [. Gallice, Schémas de Roe génralisés : Positivité et entropie, Rapport MAB 00009, Schémas de type Godunov entropiques et positifs préservant les discontinuités de contact, pp.149-152, 2000.

G. Gallice, Solveurs simples positifs et entropiques pour les syst??mes hyperboliques avec terme source, Comptes Rendus Mathematique, vol.334, issue.8, pp.1-4, 2002.
DOI : 10.1016/S1631-073X(02)02307-5

G. Gallice, Schémas équilibre et Solveurs de Riemann simples entropiques pour des systèmes hyperboliques avec terme source : application à la Dynamique des Gaz avec gravité, Rapport LRC-MAB, p.5, 2002.

G. Gallice, Un solveur de type Roe pour le système de Powell pour la MHD, Rapport LRC-MAB, p.9, 2002.

G. Gallice, Roe's Matrices for General Conservation Laws in Eulerian or Lagrangian Coordinates. Application to Gas Dynamics and MHD, C. R. Acad. Sci. Paris, vol.321, pp.1069-1072, 1995.

G. Gallice, Simple Riemann's Godunov-type's Scheme for hyperbolic non-conservative Systems and Construction of a positive and entropic Godunov-type's Scheme for the Powell System in multidimensional MHD, in preparation, GAL10] G. Gallice, Solveurs simples positifs et entropique pour les systèmes hyperboliques avec terme source, pp.713-716, 2002.

. Gall-]-t, J. Gallouet, N. Hérard, and . Seguin, Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography, 2001.

A. [. Greenberg and . Leroux, A Well Balanced Scheme designed for Inhomogeneous Scalar Conservation Laws, SIAM J. Numer. Anal, vol.331, pp.1-16, 1996.

J. M. Greenberg, A. Y. 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.34-39, 1997.
DOI : 10.1137/S0036142995286751

M. [. Garcia-navarro and . Vazquez-cendon, On numerical treatment of the source terms in the shallow water equations, Computers & Fluids, vol.29, issue.8, pp.29-37, 2000.
DOI : 10.1016/S0045-7930(99)00038-9

. K. Godu-]-s and . Godunov, Dierence Methods for the Numerical Calculation of the Equations of Fluid Dynamics, Math. Sb, vol.47, pp.271-306, 1959.

. [. Godunov, Résolution numérique des problèmes multimensionnels de la dynamique des gaz, 1979.

J. Gre-]-gressier, Robustesse et précision des schémas décentrés pour les écoulements compressibles High Reolution Schemes for Hyperbolic Conservation Laws, ENSAE J. Comp. Phys, vol.49, pp.357-393, 1983.

. A. Hll, P. D. Harten, and B. Lax, Van Leer, On Upstream Dierencing and Godunov-type Schemes for Hyperbolic Conservation Laws, Icase Report, pp.82-87, 1982.

. B. Hlle, C. D. Einfeldt, P. L. Munz, B. Roe, and . Sjogreen, On Godunov-Type Methods near Low Densities, SIAM J. Numer. Anal, vol.25, issue.2, pp.294-318, 1988.

. Y. Hou-]-t, P. L. Hou, and . Floch, 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

. A. Hy, M. Harten, P. D. Hyman, and . Lax, On nite dierence approximations and entropy conditions for shocks, Comm. Pure Appli. Math, vol.29, pp.297-322, 1976.

P. Jan-]-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

S. Jin and . Steady, 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

R. Khanfir-]-khanr, B. Kh-pe-]-khobalatte, and B. Perthame, Approximation volumes nis de type cinétique du système hyperbolique de la MHD idéale compressible à pression isotrope, Maximum Principle on the Entropy and Minimal Limitations for Kinetic Schemes, pp.119-132, 1994.

P. and L. Floch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Communications in Partial Differential Equations, vol.2, issue.6, pp.669-727, 1988.
DOI : 10.1016/0022-0396(87)90188-4

[. Floch and T. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Mathematicum, vol.5, issue.5, p.254, 1992.
DOI : 10.1515/form.1993.5.261

. A. Ler, A. Y. Chinnayya, and . Leroux, A New General Riemann solver for the shallow-water equations with friction and topography, Preprint, 1999.

[. 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 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

T. Linde, A Three-Dimensional Adaptative Multiuid MHD Model of the Heliosphere, Sequel to AUSM: AUSM +, J. Comp. Phys, pp.364-382, 1996.

M. S. Liou, On a new class of ux-splitting, 13 th Int, Conf. on Num. Meth. in uid dynamics, pp.115-119, 1992.

G. Mehl-]-mehlman, Etudes de quelques problèmes liés aux écoulements en déséquilibre chimique et thermique On Godunov-type schemes for lagrangian gas dynamics, Ecole Polytechnique SIAM J. Numer. Anal, pp.31-48, 1991.

A. Dedner, F. Kemm, D. Kroner, C. D. 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

S. Osher-]-osher and F. Solomon, Upwind Dierence Schemes for Hyperbolic Systems of Conservation Laws, Math. Comp, vol.38, pp.321-374, 1982.

B. Perthame, Boltzmann Type Schemes for Gas Dynamics and the Entropy Property, SIAM Journal on Numerical Analysis, vol.27, issue.6, pp.191-205, 1989.
DOI : 10.1137/0727081

R. Botchorishvili, B. Perthame, and A. Vasseur, Equilibrium Schemes for Scalar Conservation Laws with Sti Sources, 2000.

E. Audusse, M. O. Bristeau, and B. Perthame, Kinetic Schemes for Saint-Venant Equations with Source Terms on Unstructured Grids, 2000.
URL : https://hal.archives-ouvertes.fr/inria-00072657

. G. Powell-]-k and . Powell, An Approximate Riemann Solver that works in more than one dimension, Rapport Icase, pp.94-118, 1994.

E. Godlewski and P. , Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol.118, 1996.
DOI : 10.1007/978-1-4612-0713-9

[. A. Raviart and L. Sainsaulieu, A NONCONSERVATIVE HYPERBOLIC SYSTEM MODELING SPRAY DYNAMICS. PART I: SOLUTION OF THE RIEMANN PROBLEM, Mathematical Models and Methods in Applied Sciences, vol.05, issue.03, pp.297-333, 1995.
DOI : 10.1142/S021820259500019X

. 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

. G. Sod and . Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of Computational Physics, vol.27, issue.1, pp.1-31, 1978.
DOI : 10.1016/0021-9991(78)90023-2

[. Steger, J. L. Warming, and R. F. , Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods, Journal of Computational Physics, vol.40, issue.2, pp.263-293, 1981.
DOI : 10.1016/0021-9991(81)90210-2

E. Tadmor, A minimum entropy principle in the gas dynamics equations, Applied Numerical Mathematics, vol.2, issue.3-5, pp.211-219, 1986.
DOI : 10.1016/0168-9274(86)90029-2

. Toro, Riemann solvers and Numerical Methods for Fluid Dynamics, 1997.

G. Toth, The ?????B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes, Journal of Computational Physics, vol.161, issue.2, pp.605-652, 1999.
DOI : 10.1006/jcph.2000.6519

I. Toumi, A weak formulation of roe's approximate riemann solver, Journal of Computational Physics, vol.102, issue.2, pp.360-373, 1992.
DOI : 10.1016/0021-9991(92)90378-C

. Vleer and B. Van-leer, Towards the Ultimate Conservative Dierence Scheme V. A Second-Order Sequel to Godunov's Method, J. Comp. Phys, vol.32, pp.101-136, 1979.

M. S. Liou, J. S. Shuen, B. Van, and . Leer, Splitting of inviscid uxes for real gases, J.C.P, vol.27, pp.1-24, 1990.

. Vil and P. Villedieu, Approximation de type cinétique du système hyperbolique de la dynamique des gaz hors-équilibre thermochimique, 1994.

. Xu and K. Xu, Gas-Kinetic Theory-Based Flux Splitting Method for Ideal Magnetohydrodynamics, Icase Report, pp.98-53
DOI : 10.1006/jcph.1999.6280

A. Harten, R. F. Warming, and H. C. Yee, Implicit total variation diminishing schemes for steady states calculations, pp.83-1902, 1983.

. [. Yee, Construction of explicit and implicit symmetric TVD schemes and their applications, Journal of Computational Physics, vol.68, issue.1, pp.151-179, 1987.
DOI : 10.1016/0021-9991(87)90049-0

[. Zachary, S. Colella, and P. , A higher-order godunov method for the equations of ideal magnetohydrodynamics, Journal of Computational Physics, vol.99, issue.2, pp.341-347, 1992.
DOI : 10.1016/0021-9991(92)90213-I

J. G. Zhou, D. M. Causon, C. G. Mingham, and A. D. Ingram, The Surface Gradient Method for the Treatment of Source Terms in the Shallow-Water Equations, Journal of Computational Physics, vol.168, issue.1, pp.1-25, 2001.
DOI : 10.1006/jcph.2000.6670