A discrete velocity numerical scheme for the two-dimensional bitemperature Euler system. (English) Zbl 1482.65158

Authors’ abstract: This paper is devoted to the numerical approximation of the bidimensional bitemperature Euler system. This model is a nonconservative hyperbolic system describing an out of equilibrium plasma in a quasi-neutral regime, with applications in inertial confinement fusion. One main difficulty here is to handle shock solutions involving the product of the velocity by pressure gradients. The authors develop a second order numerical scheme by using a discrete BGK relaxation model. The second order extension is based on a subdivision of each cartesian cell into four triangles to perform affine reconstructions of the solution. Such ideas have been developed in the literature for systems of conservation laws. Here, it is shown how they can be used in a nonconservative setting. The numerical method is implemented and tested in the last part of the paper.


65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N08 Finite volume methods for boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76L05 Shock waves and blast waves in fluid mechanics
35Q31 Euler equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D75 Nuclear reactor theory; neutron transport


Full Text: DOI


[1] R. Abgrall and S. Karni, A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), pp. 2759-2763. · Zbl 1188.65134
[2] D. Aregba-Driollet and S. Brull, About viscous approximations of the bitemperature Euler system, Commun. Math Sci., 17 (2019), pp. 1135-1147. · Zbl 1428.82057
[3] D. Aregba-Driollet and S. Brull, Modelling and Numerical Study of the Polyatomic Bitemperature Euler System, in revision. · Zbl 1407.82044
[4] D. Aregba-Driollet, S. Brull, and X. Lhébrard, Nonconservative hyperbolic systems in fluid mechanics, in SMAI 2017-8ème Biennale Française des Mathématiques Appliquées et Industrielles, ESAIM Proc. Surveys 64, EDP Sciences, Les Ulis, 2018. · Zbl 1461.76377
[5] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for systems of conservation laws, in Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, Internat. Ser. Numer. Math. 129, Birkhäuser, Basel, 1999, pp. 1-10. · Zbl 0930.65095
[6] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal., 37 (2000), pp. 1973-2004. · Zbl 0964.65096
[7] D. Aregba-Driollet, R. Natalini, and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Math. Comp., 73 (2004), pp. 63-94. · Zbl 1031.65093
[8] D. Aregba–Driollet, J. Breil, S. Brull, B. Dubroca, and E. Estibals, Modelling and numerical approximation for the nonconservative bitemperature Euler model, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 1353-1383. · Zbl 1417.65158
[9] C. Berthon, Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys., 218 (2006), pp. 495-509. · Zbl 1161.65345
[10] F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Stat. Phys., 95 (1999), pp. 113-170. · Zbl 0957.82028
[11] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birknäuser Verlag, Basel, 2004. · Zbl 1086.65091
[12] S. Brull, B. Dubroca, and X. Lhébrard, Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field, Comput. Fluids, 14 (2021). · Zbl 07352784
[13] S. Brull, B. Dubroca, and C. Prigent, A kinetic approach of the bi-temperature Euler model, Kinet. Relat. Models, 13 (2020), pp. 33-61. · Zbl 1434.82033
[14] C. Calgaro, E. Creusé, T. Goudon, and Y. Penel, Positivity-preserving schemes for Euler equations: Sharp and practical CFL conditions, J. Comput. Phys., 234 (2013), pp. 417-438. · Zbl 1284.65110
[15] C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressurelaws and explicit predictor-corrector schemes, Numer. Math., 103 (2005), pp. 451-478. · Zbl 1136.76395
[16] G.-Q. Chen, C. Levermore, and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47 (1994), pp. 787-830. · Zbl 0806.35112
[17] F. Coquel and C. Marmignon, Numerical methods for weakly ionized gas, Astrophys. Space Sci., 260 (1998), pp. 15-27. · Zbl 0962.76571
[18] G. Dal Maso, P. Le Floch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), pp. 483-548. · Zbl 0853.35068
[19] B. Einfeldt, C.-D. Munz, P. L. Roe, and B. Sjögreen, On Godunov-type methods near low densities, J. Comput. Phys., 92 (1991), pp. 273-295. · Zbl 0709.76102
[20] E. Estibals, H. Guillard, and A. Sangam, Bi-temperature Euler Equations Modeling for Fusion Plasma, Tech. Report RR-9026, INRIA Sophia-Antipolis, 2017.
[21] J. D. Huba and B. P. Branch, Revised Nrl Plasma Formulary, Naval Research Laboratory, Washington, D.C., 2004.
[22] T.-P. Liu, Hyperbolic conservation laws with relaxation, Commun. Math. Phys., 108 (1987), pp. 153-175. · Zbl 0633.35049
[23] R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Differential Equations, 148 (1998), pp. 292-317. · Zbl 0911.35073
[24] C. Parés, Path-conservative numerical methods for nonconservative hyperbolic systems, in Numerical Methods for Balance Laws, Quad. Mat. 24, Department of Mathematics, Seconda University Napoli, Caserta, 2009, pp. 67-121. · Zbl 1266.65148
[25] B. Perthame and Y. Qiu, A variant of van Leer’s method for multidimensional systems of conservation laws, J. Comput. Phys., 112 (1994), pp. 370-381. · Zbl 0816.65055
[26] B. Perthame and C.-W. Shu, On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73 (1996), pp. 119-130. · Zbl 0857.76062
[27] B. Perthame and C.-W. Shu, On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73 (1996), pp. 119-130. · Zbl 0857.76062
[28] D. Serre, Relaxations semi-linéaire et cinétique des systèmes de lois de conservation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), pp. 169-192. · Zbl 0963.35117
[29] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, Berlin, 2009. · Zbl 1227.76006
[30] Q. Wargnier, S. Faure, B. Graille, T. Magin, and M. Massot, Numerical treatment of the nonconservative product in a multiscale fluid model for plasmas in thermal nonequilibrium: application to solar physics, SIAM J. Sci. Comput., 42 (2020), pp. 1-27. · Zbl 1439.82064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.