# Exercices corrigés d’algèbre linéaire by Damien Etienne

By Damien Etienne

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Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete Contin. Dyn. Syst. Ser. B 6(5), 1027–1050 (2006) 8. : Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete Contin. Dyn. Syst. Ser. B 3(1), 1–20 (2003) 9. : Asymptotic l 1 -decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49(1), 113–142 (2000) 10. : Entropy-dissipative discretization of nonlinear diffusion equations and discrete beckner inequalities.

2 (3) The flux of the conservative system is given by F(W ) = (ρu, ρu 2 + p, (ρe + p)u, ρϕu)T . (4) Interpolated Pressure Laws 39 The pressure law is of the form p = P(ρ, ε, ϕ). (5) If at the initial time t = 0 the mass fraction ϕ takes only two values 0 (pure liquid phase) and 1 (pure gas phase) then it is also true at any later time. This property implies that theoretically it is only necessary to provide the pressure laws P(ρ, ε, 0) for the liquid and P(ρ, ε, 1) for the gas. A classic choice is the stiffened gas pressure law P(ρ, ε, ϕ) = (γ (ϕ) − 1)ρε − γ (ϕ)π(ϕ), (6) γ (1) = γ1 > 1, π(1) = π1 = 0, γ (0) = γ2 > 1, π(0) = π2 > 0.

Fr/hal-00924282 11. : Finite volume scheme for multi-dimensional driftdiffusion equations and convergence analysis. m2an. Math. Model. Numer. Anal. 37(2), 319– 338 (2003) 12. : Asymptotic behavior of the Scharfetter-Gummel scheme for the drift-diffusion model. In: Finite volumes for complex applications. VI. Problems and perspectives. Volume 1, 2, Springer Proceedings Mathematics, vol. 4, pp. 235–243. Springer, Heidelberg (2011) 13. : Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations.