By Jürgen Fuhrmann, Mario Ohlberger, Christian Rohde

The first quantity of the court cases of the seventh convention on "Finite Volumes for advanced functions" (Berlin, June 2014) covers issues that come with convergence and balance research, in addition to investigations of those equipment from the viewpoint of compatibility with actual rules. It collects jointly the targeted invited papers, in addition to the reviewed contributions from the world over prime researchers within the box of research of finite quantity and similar equipment. Altogether, a slightly entire evaluate is given of the state-of-the-art within the field.

The finite quantity technique in its a number of types is an area discretization method for partial differential equations in keeping with the basic actual precept of conservation. contemporary a long time have introduced major good fortune within the theoretical knowing of the strategy. Many finite quantity equipment shield extra qualitative or asymptotic homes, together with greatest rules, dissipativity, monotone decay of loose power, and asymptotic balance. as a result of those houses, finite quantity equipment belong to the broader category of suitable discretization equipment, which protect qualitative houses of continuing difficulties on the discrete point. This structural method of the discretization of partial differential equations turns into really very important for multiphysics and multiscale purposes.

Researchers, PhD and masters point scholars in numerical research, clinical computing and comparable fields akin to partial differential equations will locate this quantity worthy, as will engineers operating in numerical modeling and simulations.

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**Extra info for Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects: FVCA 7, Berlin, June 2014 (Springer Proceedings in Mathematics & Statistics)**

**Example text**

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.