I. Aavatsmark, T. Barkve, O. Boe, and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anistropic media. Part I: derivation of the methods

I. Aavatsmark, T. Barkve, O. Boe, and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anistropic media. Part II: discussion and numerical results

I. Aavatsmark, G. T. Eigestad, and R. A. Klausen, Numerical Convergence of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions, 2006.
DOI : 10.1007/0-387-38034-5_1

I. Aavvatsmark, G. T. Eigestad, R. A. Klausen, M. F. Wheeler, and I. Yotov, Convergence of a symmetric MPFA method on quadrilateral grids, Computational Geosciences, vol.44, issue.4, 2005.
DOI : 10.1007/s10596-007-9056-8

R. Abgrall, R. Loubère, and J. Ovadia, A Lagrangian Discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, Int. J. Numer. Meth. Fluids, vol.44, pp.645-663, 2004.

F. L. Adessio, J. K. Baumgardner, N. L. Dukowicz, B. A. Johnson, R. M. Kashiwa et al., Caveat: a computer code for fluid dynamics problems with large distortion and internal slip, p.905, 1992.

F. L. Adessio, D. E. Carroll, J. K. Dukowicz, N. L. Johnson, B. A. Kashiwa et al., Caveat: a computer code for fluid dynamics problems with large distortion and internal slip, 1986.

. Atzeni, The physical basis for numerical fluid simulations in laser fusion, Plasma Physics and Controlled Fusion, vol.29, issue.11, pp.1535-1604, 1987.
DOI : 10.1088/0741-3335/29/11/001

J. Atzeni, L. Davies, J. J. Hallo, P. Honrubia, M. Maire et al., Studies on targets for inertial fusion ignition demonstration at the HiPER facility, Nuclear Fusion, vol.49, issue.5, p.49, 2009.
DOI : 10.1088/0029-5515/49/5/055008

S. Atzeni and J. Meyer-ter-vehn, The physics of inertial fusion, Oxford Science publications, 2004.
DOI : 10.1093/acprof:oso/9780198562641.001.0001

A. Barlow, A cell centred lagrangian godunov scheme for shock hydrodynamics. Computers and Fluids, 2010.

A. Barlow, D. Burton, and M. Shashkov, Compatible, Energy and Symmetry Preserving 2D Lagrangian Hydrodynamics in rz-Cylindrical Coordinates, Procedia Computer Science, vol.1, pp.1887-1895, 2010.

A. J. Barlow, A compatible finite element multi-material ALE hydrodynamics algorithm, International Journal for Numerical Methods in Fluids, vol.141, issue.8, pp.953-964, 2008.
DOI : 10.1002/fld.1593

T. J. Barth, Numerical methods for conservation laws on structured and unstructured meshes, 2003.

T. J. Barth and D. C. Jespersen, The design and application of upwind schemes on unstructured meshes, 27th Aerospace Sciences Meeting, 1989.
DOI : 10.2514/6.1989-366

A. L. Bauer, D. E. Burton, E. J. Caramana, R. Loubère, M. J. Shashkov et al., The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics, Journal of Computational Physics, vol.218, issue.2, pp.572-593, 2006.
DOI : 10.1016/j.jcp.2006.02.024

M. Ben-artzi and A. Birman, Application of the ???Generalized Riemann Problem??? method to 1-D compressible flows with material interfaces, Journal of Computational Physics, vol.65, issue.1, pp.170-178, 1986.
DOI : 10.1016/0021-9991(86)90010-0

M. Ben-artzi and J. Falcovitz, A second-order Godunov-type scheme for compressible fluid dynamics, Journal of Computational Physics, vol.55, issue.1, pp.1-32, 1984.
DOI : 10.1016/0021-9991(84)90013-5

M. Ben-artzi and J. Falcovitz, An Upwind Second-Order Scheme for Compressible Duct Flows, SIAM Journal on Scientific and Statistical Computing, vol.7, issue.3, pp.744-768, 1986.
DOI : 10.1137/0907051

M. Ben-artzi and J. Falcovitz, Generalized Riemann problems in Computational Fluids Dynamics, 2003.
DOI : 10.1017/CBO9780511546785

M. Ben-artzi, J. Li, and G. Warnecke, A direct Eulerian GRP scheme for compressible fluid flows, Journal of Computational Physics, vol.218, issue.1, pp.19-43, 2006.
DOI : 10.1016/j.jcp.2006.01.044

D. J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Computer Methods in Applied Mechanics and Engineering, vol.99, issue.2-3, pp.235-394, 1992.
DOI : 10.1016/0045-7825(92)90042-I

J. Botsis and M. Deville, Mécanique des milieux continus, Presses Polytechniques et Universitaires Romandes, 2006.

A. F. Bower, Applied Mechanics of Solids, 2010.

S. I. Braginskii, Transport Process in a Plasma, In Reviews of Plasma Physics Consultants Bureau, vol.I, pp.205-311, 1965.

J. Breil, L. Hallo, P. Maire, and M. Olazabal-loumé, Hydrodynamic instabilities in axisymmetric geometry self-similar models and numerical simulations, Laser and Particle Beams, vol.1, issue.02, pp.155-160, 2005.
DOI : 10.1017/S026303460321301X__S026303460321301X

J. Breil and P. Maire, A cell-centered diffusion scheme on two-dimensional unstructured meshes, Journal of Computational Physics, vol.224, issue.2, pp.785-823, 2007.
DOI : 10.1016/j.jcp.2006.10.025

J. Breil, P. Maire, . Ph, G. Nicola¨?nicola¨?, and . Schurtz, Modelling of the magnetic field effects in hydrodynamic codes using a second order tensorial diffusion scheme, Journal of Physics: Conference Series The fifth International Conference on Inertial Fusion Sciences and Applications, pp.1742-6596, 2007.
DOI : 10.1088/1742-6596/112/2/022035

A. Burbeau-augoula, A Node-Centered Artificial Viscosity Method for Two-Dimensional Lagrangian Hydrodynamics Calculations on a Staggered Grid, Communications in Computational Physics, vol.8, pp.877-900, 2009.
DOI : 10.4208/cicp.030709.161209a

D. E. Burton, Multidimensional Discretization of Conservation Laws for Unstructured Polyhedral Grids, 1994.

J. C. Campbell and M. J. Shashkov, A Tensor Artificial Viscosity Using a Mimetic Finite Difference Algorithm, Journal of Computational Physics, vol.172, issue.2, pp.739-765, 2001.
DOI : 10.1006/jcph.2001.6856

J. C. Campbell and M. J. Shashkov, A compatible Lagrangian hydrodynamics algorithm for unstructured grids, Selçuk J. Appl. Math, vol.4, issue.2, pp.53-70, 2003.

E. J. Caramana, M. J. Shashkov, and P. P. Whalen, Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations, Journal of Computational Physics, vol.144, issue.1, pp.70-97, 1998.
DOI : 10.1006/jcph.1998.5989

E. J. Caramana, D. E. Burton, M. J. Shashkov, and P. P. Whalen, The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy, Journal of Computational Physics, vol.146, issue.1, pp.227-262, 1998.
DOI : 10.1006/jcph.1998.6029

E. J. Caramana and M. J. Shashkov, Elimination of Artificial Grid Distortion and Hourglass-Type Motions by Means of Lagrangian Subzonal Masses and Pressures, Journal of Computational Physics, vol.142, issue.2, pp.521-561, 1998.
DOI : 10.1006/jcph.1998.5952

E. J. Caramana and P. Whalen, Numerical Preservation of Symmetry Properties of Continuum Problems, Journal of Computational Physics, vol.141, issue.2, pp.174-198, 1998.
DOI : 10.1006/jcph.1998.5912

G. Carré, S. Delpino, B. Després, and E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, Journal of Computational Physics, vol.228, issue.14, pp.5160-5183, 2009.
DOI : 10.1016/j.jcp.2009.04.015

C. E. Castro and E. F. Toro, Solvers for the high-order Riemann problem for hyperbolic balance laws, Journal of Computational Physics, vol.227, issue.4, pp.2481-2513, 2008.
DOI : 10.1016/j.jcp.2007.11.013

R. Chéret, Détonique des Explosifs Condensés, volume I, 1988.

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Archive for Rational Mechanics and Analysis, vol.4, issue.1, pp.167-178, 1963.
DOI : 10.1007/BF01262690

P. I. Crumpton, G. J. Shaw, and A. F. Ware, Discretisation and Multigrid Solution of Elliptic Equations with Mixed Derivative Terms and Strongly Discontinuous Coefficients, Journal of Computational Physics, vol.116, issue.2, pp.343-358, 1995.
DOI : 10.1006/jcph.1995.1032

R. Dautray and J. Watteau, La fusion thermonucléaire par laser, volume II, Eyrolles, 1991.

L. Davison, Fundamentals of Shock Wave Propagation in Solids, 2008.

S. R. De-groot and P. Mazur, Non-equilibrium thermodynamics, 1984.

B. Després, Lagrangian systems of conservation laws, Numerische Mathematik, vol.89, issue.1, pp.99-134, 2001.
DOI : 10.1007/PL00005465

B. Després, Lois de Conservation Euleriennes, Lagrangiennes et méthodes numériques, 2010.
DOI : 10.1007/978-3-642-11657-5

B. Després, Weak consistency of the cell-centered Lagrangian GLACE scheme on general meshes in any dimension, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.41-44, pp.2669-2679, 2010.
DOI : 10.1016/j.cma.2010.05.010

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

V. A. Dobrev, T. E. Ellis, T. V. Kolev, and R. N. Rieben, Curvilinear finite elements for Lagrangian hydrodynamics, International Journal for Numerical Methods in Fluids, vol.72, issue.1, 2010.
DOI : 10.1002/fld.2366

J. Donea, A. Huerta, J. Ph, A. Ponthot, and . Rodriguez-ferran, Encyclopedia of Computational Mechanics, chapter 14: Arbitrary Lagrangian-Eulerian methods, 2004.

R. P. Drake, High-energy-density physics, Physics Today, vol.63, issue.6, 1986.
DOI : 10.1063/1.3455249

J. K. Dukowicz, A general, non-iterative Riemann solver for Godunov's method, Journal of Computational Physics, vol.61, issue.1, pp.119-137, 1984.
DOI : 10.1016/0021-9991(85)90064-6

J. K. Dukowicz and B. Meltz, Vorticity errors in multidimensional lagrangian codes, Journal of Computational Physics, vol.99, issue.1, pp.115-134, 1992.
DOI : 10.1016/0021-9991(92)90280-C

M. G. Edwards and C. F. Rogers, Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Computational Geosciences, vol.2, issue.4, pp.259-290, 1998.
DOI : 10.1023/A:1011510505406

S. Atzeni, Studies on targets for inertial fusion ignition demonstration at the HiPER facility, Nuclear Fusion, vol.49, issue.5, 2009.
DOI : 10.1088/0029-5515/49/5/055008

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume methods. Handbook of Numerical Analysis, 2000.
URL : https://hal.archives-ouvertes.fr/hal-00346077

S. Galera, P. Maire, and J. Breil, A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction, Journal of Computational Physics, vol.229, issue.16, pp.5755-5787, 2010.
DOI : 10.1016/j.jcp.2010.04.019
URL : https://hal.archives-ouvertes.fr/inria-00453534

G. Gallice, Approximation numérique de systèmes hyperboliques non-linéaires conservatifs ou non-conservatifs. HabilitationàHabilitationà Diriger des Recherches, 2002.

G. Gallice, Positive and Entropy Stable Godunov-type Schemes for Gas Dynamics and MHD Equations in Lagrangian or Eulerian Coordinates, Numerische Mathematik, vol.94, issue.4, pp.673-713, 2003.
DOI : 10.1007/s00211-002-0430-0

R. V. Garimella and K. Lipnikov, Solution of the diffusion equation in multi-material domains by sub-division of elements along reconstructed interfaces, International Journal for Numerical Methods in Fluids, vol.227, issue.11, 2010.
DOI : 10.1002/fld.2350

P. Germain, Mécanique, volume I. Ellipses, 1986.

W. B. Goad, WAT: A Numerical Method for Two-Dimensional Unsteady Fluid Flow, 1960.

E. Godlewski and P. Raviart, Hyperbolic Systems of Conservation Laws, 2000.
URL : https://hal.archives-ouvertes.fr/hal-00113734

V. Gyrya and K. Lipnikov, High-order mimetic finite difference method for diffusion problems on polygonal meshes, Journal of Computational Physics, vol.227, issue.20, pp.8841-8854, 2008.
DOI : 10.1016/j.jcp.2008.06.028

L. Hallo, M. Olazabal-loumé, X. Ribeyre, V. Dréan, G. Schurtz et al., Hydrodynamic and symmetry safety factors of hiper's targets. Plasma Phys, Control. Fusion, p.51, 2009.

F. Hermeline, A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes, Journal of Computational Physics, vol.160, issue.2, pp.481-499, 2000.
DOI : 10.1006/jcph.2000.6466

F. Hermeline, A finite volume method for approximating 3D diffusion operators on general meshes, Journal of Computational Physics, vol.228, issue.16, pp.5763-5786, 2009.
DOI : 10.1016/j.jcp.2009.05.002

C. W. Hirt, A. Amsden, and J. L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, vol.14, issue.3, pp.227-253, 1974.
DOI : 10.1016/0021-9991(74)90051-5

W. H. Hui, P. Y. Li, and Z. W. Li, A Unified Coordinate System for Solving the Two-Dimensional Euler Equations, Journal of Computational Physics, vol.153, issue.2, pp.596-637, 1999.
DOI : 10.1006/jcph.1999.6295

J. Hyman, J. E. Morel, M. Shashkov, and S. Steinberg, Mimetic finite difference methods for diffusion equations, Computational Geosciences, vol.6, issue.3/4, pp.333-352, 2002.
DOI : 10.1023/A:1021282912658

J. R. Kamm and F. X. Timmes, On efficient generation of numerically robust Sedov solutions, 2007.

D. S. Kershaw, Differencing of the diffusion equation in Lagrangian hydrodynamic codes, Journal of Computational Physics, vol.39, issue.2, pp.375-395, 1981.
DOI : 10.1016/0021-9991(81)90158-3

R. E. Kidder, Laser-driven compression of hollow shells: power requirements and stability limitations, Nuclear Fusion, vol.16, issue.1, pp.3-14, 1976.
DOI : 10.1088/0029-5515/16/1/001

R. A. Klausen and T. F. Russell, Relationships among some locally conservative discretization methods which handle discontinuous coefficients, Computational Geosciences, vol.6, issue.4, pp.341-377, 2004.
DOI : 10.1007/s10596-005-1815-9

. V. Tz, R. N. Kolev, and . Rieben, A tensor artificial viscosity using a finite element approach, J. Comp. Phys, vol.228, issue.22, pp.8336-8366, 2010.

V. P. Kolgan, Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics, Journal of Computational Physics, vol.230, issue.7
DOI : 10.1016/j.jcp.2010.12.033

Y. Kuznetsov, K. Lipnikov, and M. Shashkov, The mimetic finite difference method on polygonal meshes for diffusion-type problems, Computational Geosciences, vol.88, issue.3/4, pp.301-324, 2004.
DOI : 10.1007/s10596-004-3771-1

P. Lascaux, Application de la méthode desélémentsdeséléments finis en hydrodynamique bidimensionnelle utilisant les variables de Lagrange, 1972.

P. Lascaux, Application of the Finite Element Method to 2D Lagrangian hydrodynamics In Finite element methods in flow problems, Proceedings of the International Symposium, pp.139-152, 1974.

P. Lascaux and R. Théodor, Analyse Numérique matricielle appliquéè a l'art de l'ingénieur, volume II, 2000.

P. Lax and B. Wendroff, Systems of conservation laws, Commun. Pur. Appl. Math, 1960.

E. Lescoute, T. De-rességuier, J. Chevalier, J. Breil, P. Maire et al., Ejection of spalled layers from laser shock-loaded metals, Journal of Applied Physics, vol.108, issue.9, 2010.
DOI : 10.1063/1.3500317
URL : https://hal.archives-ouvertes.fr/hal-01136321

E. Lescoute, T. De-rességuier, J. Chevalier, J. Breil, P. Maire et al., Experimental and numerical study of dynamic fragmentation in laser shock-loaded gold and aluminium targets, Computers Materials and Continua, 2010.

J. Li and Z. Sun, Remark on the generalized Riemann problem method for compressible fluid flows, Journal of Computational Physics, vol.222, issue.2, pp.796-808, 2007.
DOI : 10.1016/j.jcp.2006.08.017

K. Lipnikov, J. E. Morel, and M. Shashkov, Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes, Journal of Computational Physics, vol.199, issue.2, pp.589-597, 2004.
DOI : 10.1016/j.jcp.2004.02.016

K. Lipnikov and M. Shashkov, A framework for developing a mimetic tensor artificial viscosity for Lagrangian hydrocodes on arbitrary polygonal meshes, Journal of Computational Physics, vol.229, issue.20, pp.7911-7941, 2010.
DOI : 10.1016/j.jcp.2010.06.045

K. Lipnikov, M. Shashkov, and D. Svyatskiy, The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes, Journal of Computational Physics, vol.211, issue.2, pp.473-491, 2006.
DOI : 10.1016/j.jcp.2005.05.028

K. Lipnikov, M. Shashkov, D. Svyatskiy, and Y. Vassilevski, Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes, Journal of Computational Physics, vol.227, issue.1, pp.492-512, 2007.
DOI : 10.1016/j.jcp.2007.08.008

K. Lipnikov, M. Shashkov, and I. Yotov, Local flux mimetic finite difference methods, Numerische Mathematik, vol.44, issue.1, 2005.
DOI : 10.1007/s00211-008-0203-5

K. Lipnikov, M. Shashkov, and I. Yotov, Local flux mimetic finite difference methods, Numerische Mathematik, vol.44, issue.1, pp.115-152, 2009.
DOI : 10.1007/s00211-008-0203-5
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.7602

K. Lipnikov, D. Svyatskiy, and Y. Vassilevski, Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes, Journal of Computational Physics, vol.228, issue.3, pp.703-716, 2009.
DOI : 10.1016/j.jcp.2008.09.031

R. Loubère, Une Méthode Particulaire Lagrangienne de type Galerkin Discontinu Applicationàcationà la Mécanique des Fluides et l'Interaction Laser/Plasma, 2002.

R. Loubère and E. J. Caramana, The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics, Journal of Computational Physics, vol.216, issue.1, pp.1-18, 2006.
DOI : 10.1016/j.jcp.2005.11.022

R. Loubère, M. Shashkov, and B. Wendroff, Volume consistency in a staggered grid Lagrangian hydrodynamics scheme, Journal of Computational Physics, vol.227, issue.8, pp.3731-3737, 2008.
DOI : 10.1016/j.jcp.2008.01.006

G. Luttwak and J. Falcovitz, Slope limiting for vectors: a novel vector limiting algorithm Conference on Numerical methods for multi-material fluid flows, 2009.

G. Luttwak and J. Falcovitz, Slope limiting for vectors: A novel vector limiting algorithm, International Journal for Numerical Methods in Fluids, vol.72, issue.5, 2010.
DOI : 10.1002/fld.2367

S. M. Murman, M. Berger, and M. J. Aftosmis, Analysis of slope limiters on irregular grids, 2005.

P. Maire, A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, Journal of Computational Physics, vol.228, issue.18, pp.6882-6915, 2009.
DOI : 10.1016/j.jcp.2009.06.018
URL : https://hal.archives-ouvertes.fr/inria-00372105

P. Maire, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, Journal of Computational Physics, vol.228, issue.7, pp.2391-2425, 2009.
DOI : 10.1016/j.jcp.2008.12.007
URL : https://hal.archives-ouvertes.fr/inria-00322369

P. Maire, A high-order one-step sub-cell force-based discretization for cell-centered lagrangian hydrodynamics on polygonal grids. Computers and Fluids, 2010.

P. Maire, A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, International Journal for Numerical Methods in Fluids, vol.61, issue.3, 2010.
DOI : 10.1002/fld.2328

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

P. Maire and J. Breil, A second-order cell-centered Lagrangian scheme for twodimensional compressible flow problems, Int. J. Numer. Meth. Fluids, issue.8, pp.561417-1423, 2008.

P. Maire, R. Loubère, and P. Vachal, Abstract, Communications in Computational Physics, vol.146, issue.04, 2010.
DOI : 10.1016/j.jcp.2005.11.022

P. Maire and B. Nkonga, Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, Journal of Computational Physics, vol.228, issue.3, pp.799-821, 2009.
DOI : 10.1016/j.jcp.2008.10.012
URL : https://hal.archives-ouvertes.fr/inria-00290717

L. Margolin, M. Shashkov, and P. Smolarkiewicz, A discrete operator calculus for finite difference approximations, Computer Methods in Applied Mechanics and Engineering, vol.187, issue.3-4, pp.365-383, 2000.
DOI : 10.1016/S0045-7825(00)80001-8

L. G. Margolin and M. J. Shashkov, Using a Curvilinear Grid to Construct Symmetry-Preserving Discretizations for Lagrangian Gas Dynamics, Journal of Computational Physics, vol.149, issue.2, pp.389-417, 1999.
DOI : 10.1006/jcph.1998.6161

L. G. Margolin, M. J. Shashkov, and M. A. Taylor, Symmetry-preserving discretizations for Lagrangian gas dynamics, Proceedings of the 3rd European Conference, Numerical Mathematics and Advanced Applications, pp.725-732, 2000.

J. E. Marsden and T. J. Hughes, Mathematical Foundations of Elasticity. Dover, 1994. [116] C. Mazeran. Sur la structure mathématique et l'approximation numérique de l'hydrodynamique Lagrangienne bidimensionelle, 2007.

R. Menikoff, Notes on Elastic-Plastic Flow, 2003.

J. E. Morel, J. E. Dendy, M. L. Hall, and S. W. White, A cell-centered Lagrangian-mesh diffusion differencing scheme, Journal of Computational Physics, vol.103, issue.2, pp.286-299, 1992.
DOI : 10.1016/0021-9991(92)90402-K

J. E. Morel, R. M. Roberts, and M. Shashkov, A Local Support-Operators Diffusion Discretization Scheme for Quadrilateralr-zMeshes, Journal of Computational Physics, vol.144, issue.1, pp.17-51, 1998.
DOI : 10.1006/jcph.1998.5981

G. A. Moses and J. Yuan, Radiation diffusion in DRACO using Kershaw difference scheme, 2003.

J. D. Moulton, T. M. Austin, M. Shashkov, and J. E. , Mimetic preconditionners for mixed discretizations of the diffusion equation, IMA " Hot Topics " Workshop: Compatible Spatial Discretizations for Partial Differential Equa- tions, 2004.

C. D. Munz, On Godunov-Type Schemes for Lagrangian Gas Dynamics, SIAM Journal on Numerical Analysis, vol.31, issue.1, pp.17-42, 1994.
DOI : 10.1137/0731002

B. Nkonga, On the conservative and accurate CFD approximations for moving meshes and moving boundaries, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.13-14, pp.13-141801, 2000.
DOI : 10.1016/S0045-7825(00)00191-2
URL : https://hal.archives-ouvertes.fr/hal-01313354

W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, Journal of Computational Physics, vol.72, issue.1, pp.78-120, 1987.
DOI : 10.1016/0021-9991(87)90074-X

W. Noll, On the Continuity of the Solid and Fluid States, Indiana University Mathematics Journal, vol.4, issue.1, pp.3-81, 1955.
DOI : 10.1512/iumj.1955.4.54001

S. and D. Pino, A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates, Comptes Rendus Mathematique, vol.348, pp.17-181027, 2010.

O. Pironneau, Optimal Shape Design for Elliptic System, 1983.

B. J. Plohr and D. H. Sharp, A conservative Eulerian formulation of the equations for elastic flow, Advances in Applied Mathematics, vol.9, issue.4, pp.481-499, 1988.
DOI : 10.1016/0196-8858(88)90025-5

L. Potier, A finite volume method for the approximation of highly anisotropic diffusion operators on unstructured meshes, Finite Volumes for Complex Applications IV, 2005.

L. Potier, Sch??ma volumes finis monotone pour des op??rateurs de diffusion fortement anisotropes sur des maillages de triangles non structur??s, Comptes Rendus Mathematique, vol.341, issue.12, pp.787-792, 2005.
DOI : 10.1016/j.crma.2005.10.010

L. Potier, Sch??ma volumes finis pour des op??rateurs de diffusion fortement anisotropes sur des maillages non structur??s, Comptes Rendus Mathematique, vol.340, issue.12, pp.921-926, 2005.
DOI : 10.1016/j.crma.2005.05.011

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, volume I, pp.67-68, 2003.

P. A. Raviart and J. M. Thomas, IntroductionàIntroductionà l'analyse numérique deséquationsdeséquations aux dérivées partielles, 1988.

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, volume I, 1963.

B. Rebourcet, An old scheme adapted to new problems: Kershaw's Finite Differences diffusion scheme adapted to new problems. Workshop on Advanced Methods for the Diffusion Equation on General Meshes, 2010.

B. Rebourcet, Comments on the filtering of numerical instabilities in Lagrangian hydrocodes Conference on Numerical methods for multi-material fluid flows; Czech Technical University in Prague on September 10 -14, 2007.

B. Rebourcet, Some remarks on Kershaw's legacy diffusion scheme Conference on Numerical methods for multi-material fluid flows; Czech Technical University in Prague on, 2007.

X. Ribeyre, . Ph, G. Nicola¨?nicola¨?, M. Schurtz, J. Olazabal-loumé et al., Compression phase study of the hiper baseline target. Plasma Phys, Control. Fusion, p.50, 2008.

R. D. Richtmyer, Taylor instability in shock acceleration of compressible fluids, Communications on Pure and Applied Mathematics, vol.13, issue.2, pp.297-319, 1960.
DOI : 10.1002/cpa.3160130207

J. Salençon, Mécanique des milieux continus, volume I, Concepts généraux, 2005.

G. Schurtz, Private communication, 2007.

G. Schurtz, S. Gary, S. Hulin, C. Chenais-popovics, J. Gauthier et al., Revisiting Nonlocal Electron-Energy Transport in Inertial-Fusion Conditions, Physical Review Letters, vol.98, issue.9, p.98, 2007.
DOI : 10.1103/PhysRevLett.98.095002

G. Scovazzi, Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.4-6, pp.966-978, 2007.
DOI : 10.1016/j.cma.2006.08.009

G. Scovazzi, M. A. Christon, T. J. Hughes, and J. N. Shadid, Stabilized shock hydrodynamics: I. A Lagrangian method, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.4-6, pp.923-966, 2007.
DOI : 10.1016/j.cma.2006.08.008

G. Scovazzi and T. J. Hugues, Lecture Notes on Continuum Mechanics on Arbitrary Moving Domains, 2007.

G. Scovazzi, E. Love, and M. J. Shashkov, Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: Theoretical framework and two-dimensional computations, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.9-12, pp.1056-1079, 2008.
DOI : 10.1016/j.cma.2007.10.002

J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik, pp.125-263, 1959.
DOI : 10.1007/978-3-642-45914-6_2

M. Shashkov, Conservative Finite-Difference Methods on General Grids, 1996.

M. Shashkov and S. Steinberg, Support-Operator Finite-Difference Algorithms for General Elliptic Problems, Journal of Computational Physics, vol.118, issue.1, pp.131-151, 1995.
DOI : 10.1006/jcph.1995.1085

M. Shashkov and S. Steinberg, Solving Diffusion Equations with Rough Coefficients in Rough Grids, Journal of Computational Physics, vol.129, issue.2, pp.383-405, 1996.
DOI : 10.1006/jcph.1996.0257

Z. J. Shen, G. W. Yuan, J. Y. Yue, and X. Z. Liu, A cell-centered Lagrangian scheme in twodimensional cylindrical geometry, Science in China Series A: Mathematics, issue.8, pp.511479-1494, 2008.

G. A. 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

A. V. Solov-'ev, M. Yu, and . Shashkov, Difference scheme for the Dirichlet particle method in cylindrical coordinates, conserving symmetry of gas-dynamical flow, Differential Equations, vol.24, issue.7, pp.817-823, 1988.

B. Swartz, Good Neighborhoods for Multidimensional Van Leer Limiting, Journal of Computational Physics, vol.154, issue.1, pp.237-241, 1999.
DOI : 10.1006/jcph.1999.6308

L. Tallec, Modélisation et calcul des milieux continus. Editions de l'Ecole Polytechnique, 2009.

J. Thomas and D. Trujillo, Mixed finite volume methods, International Journal for Numerical Methods in Engineering, vol.25, issue.9, pp.1351-1366, 1999.
DOI : 10.1002/(SICI)1097-0207(19991130)46:9<1351::AID-NME702>3.0.CO;2-0
URL : https://hal.archives-ouvertes.fr/inria-00343041

V. T. Tikhonchuk and K. Mima, Alternative schemes for the inertial fusion energy, Fusion Engineering and Design, vol.86, issue.6-8, 2011.
DOI : 10.1016/j.fusengdes.2011.01.007

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

B. Van-leer, A historical oversight: Vladimir P. Kolgan and his high-resolution scheme, Journal of Computational Physics, vol.230, issue.7, 2010.
DOI : 10.1016/j.jcp.2010.12.032

R. S. Varga, Matrix Iterative Analysis, pp.90-91, 2000.
DOI : 10.1007/978-3-642-05156-2
URL : http://hdl.handle.net/2027/mdp.39015000491780

V. Venkatakrishnan, Convergence to Steady State Solutions of the Euler Equations on Unstructured Grids with Limiters, Journal of Computational Physics, vol.118, issue.1, pp.120-130, 1995.
DOI : 10.1006/jcph.1995.1084

F. Vilar, P. Maire, and R. Abgrall, Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for onedimensional Lagrangian hydrodynamics, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00538165

J. Von-neumann and R. D. Richtmyer, A Method for the Numerical Calculation of Hydrodynamic Shocks, Journal of Applied Physics, vol.21, issue.3, pp.232-238, 1950.
DOI : 10.1063/1.1699639

P. Whalen, Algebraic Limitations on Two-Dimensional Hydrodynamics Simulations, Journal of Computational Physics, vol.124, issue.1, pp.46-54, 1996.
DOI : 10.1006/jcph.1996.0043

Y. Yang, Q. Zhang, and D. H. Sharp, Small amplitude theory of Richtmyer???Meshkov instability, Physics of Fluids, vol.6, issue.5, pp.1856-1873, 1994.
DOI : 10.1063/1.868245

Y. B. Zel-'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, volume I, 1967.

Y. B. Zel-'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, volume II, 1967.