The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section

Abstract

Abstract. We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl. 74 (1995) 483-548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class of numerical approximations for this system. Mathematics Subject Classification. 35L65, 76N10, 76L05.

A two-fluid hyperbolic model in a porous medium

“…The proof is also almost the same as the one given in [29] in the case of Euler equations with perfect gas EOS, in a one-dimensional framework, where authors examine the particular case of flows in variable cross section ducts. It occurs in fact in the proof that, though the present system is indeed much more complex than the one examined in [29], both phases almost "decouple" through the interface, since the void fraction is one among the seven Riemann invariants of the standing wave associated with λ 0 (see Sect.…”

“…It occurs in fact in the proof that, though the present system is indeed much more complex than the one examined in [29], both phases almost "decouple" through the interface, since the void fraction is one among the seven Riemann invariants of the standing wave associated with λ 0 (see Sect. 2).…”

“…The expected rates of convergence are retrieved, focusing on so-called first-order schemes, and the nice behaviour of the discrete cell-values of Riemann invariants of the standing wave around the free/porous interface renders the scheme quite appealing. We recall that this scheme is inspired by the scheme introduced in [28] for Euler equations in variable cross section ducts (see [29] too). It also means that Property 6 is far from being sufficient, and that the dynamical well-balanced Property 7 enjoyed by W BR, the continuous counterpart of which is Property 3, seems mandatory to obtain convergence towards the correct solution.…”

A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

“…In addition, the discretization of nonconservative hyperbolic systems and of systems with source terms attracted a lot of attention in recent years. We refer to [5,6,7,16,18] for a single conservation law with source term and to [22,23,25,24] for fluid flows in a nozzle with variable cross-section. Well-balanced schemes for multi-phase flows and other models were studied in [2,9,36,38].…”

Well-Balanced Roe-Type Numerical Scheme for a Model of Two-Phase Compressible Flows

“…Besides, numerical well-balanced schemes for a single conservation law with a source term are presented in [3,5,6,13,14]. In [18,19] a well-balanced scheme for the model of fluid flows in a nozzle with variable cross-section was were built and studied. Numerical treatments of source terms for shallow water equations were constructed in [3,8,15,23,32].…”