Abstract
We sum up some new results concerning the mathematical structure of Lagrangian systems of conservation laws. These results include the assumption of Galilean invariance in the hypothesis and extend the classical symmetrization theorem of S. K. Godunov. Based on this representation theorem it is straightforward to derive numerical schemes which are non linearly stable, due to an entropy inequality satisfied by the scheme. We present a numerical application to an ICF-like calculation : the so-called Ti,Te model for a ionized gas with ionic and electronic temperatures.
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References
Bazarov I (1989). Thermodynamique. Edition MIR (in french).
Bezard F and Després B (1999). An entropic solver for ideal Lagrangian Magnetohydro-dynamics. Jour, of Comp. Phys..
Callen H B (1985). Thermodynamics and introduction to thermostatistics. John WILEY & SONS.
Coquel F (1998). Schémas hyperboliques. INRIA-France-1998.
Coquel F and Perthame B (1998). Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal., 35(6):2223–2249.
Després B (Submitted to Numer. Math.). Structure of lagrangian systems of conservation laws, approximate Riemann solvers and the entropy condition .
Després B (1997).Inégalité entropique pour un solveur conservatif du systéme de la dynamique des gaz en coordonnées de Lagrange. C. R. Acad. Sci., Série I, 324:1301–1306.
Després B (1998).Entropy inequality for high order discontinuous Galerkin approximation of Euler equations. VII conference on hyperbolic problems, ETHZ-Zurich.
Després B (1999).Discontinuous Galerkin Method for the numerical solution of Euler equations in axisymmetric geometry. DGM International Symposium, Newport.
Després B (1999). Structure des systèmes de lois de conservation en variables Lagrangiennes . Comptes Rendus de VAcadémie des Sciences, Série I, 328:721–724.
Gallice G (1995).Résolution numérique des équations de la magnétohydrodynamique idéale. In GdR SPARCH, editor, Proceedings of the conference. Méthodes numériques pour la MHD.
Godlewski E and Raviart P A (1996). Numerical approximation of hyperbolic systems of conservation laws. Number 118. Springer Verlag New York. AMS 118.
Godunov S K (1986). Lois de conservation et integrales d’énergie des equations hyperboliques. In Nonlinear hyperbolic problems, pages 135–149. Springer-Verlag. Lecture notes in mathematics.
Godunov S K (1999). Reminiscences about difference schemes. Journal of Computational Physics, (153):6–25.
Godunov S K (1959). A difference scheme for numerical computation of discontinuous solutions of equations of fluid dynamics. Mat. Sb., 89:271–306.
Godunov S K (1960). Sur la notion de solution généralisée. DAN, 134:1279–1282.
Harten A, Hyman J M, and Lax P D (1976). On finite difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math., 29:297–322.
Jaouen S (1997). Solveur entropique d’ordre élevé pour les équations de l’hydrodynamique à deux températures. Technical report, Université de Bordeaux. Mémoire de fin d’étude.
Van Leer B (1982). Flux-vector splitting for the euler equations. Technical Report 82–30, ICASE.
Munz D (1994). On Goudounov type schemes for Lagrangian formulations. SIAM Journal of Numer. Anal., 31:17–42.
Von Neumann J and Richtmyer R D (1950). A method for the numerical calculations of hydrodynamics shocks. Journal of Applied Physics, 21:232.
Noh W F (1987). Errors for calculations of strong schocks using an artificial viscosity and an artificial flux. Journal of Computational Physics, 72:78–120.
Powell K G, Roe P L, and Myong R S (1995).An upwind scheme for Magnetohy-drodynamics.In GdR SPARCH, editor, Proceedings of the conference. Méthodes numériques pour la MHD.
Richtmyer R D and Morton K W (1957). Difference methods for initial-value problems.Interscience Publishers.
Roe P L (1981). Approximate Riemann solver, parameter vectors and difference schemes.Journal of Computational Physics, 43:357–372.
Toro E F (1997). Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag.
ZePdovitch Y B and Raizer Y P (1967). Physics of shock waves and High-temperature hydrodynamic phenomena. Academic Press.
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Després, B. (2001). Lagrangian Systems of Conservation Laws and Approximate Riemann Solvers. In: Toro, E.F. (eds) Godunov Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-0663-8_25
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DOI: https://doi.org/10.1007/978-1-4615-0663-8_25
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