Coupling of general Lagrangian systems

Abstract

Abstract. This work is devoted to the coupling of two fluid models, such as two Euler systems in Lagrangian coordinates, at a fixed interface. We define coupling conditions which can be expressed in terms of continuity of some well chosen variables and then solve the coupled Riemann problem. In the present setting where the interface is characteristic, a particular choice of dependent variables which are transmitted ensures the overall conservativity.

Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws

“…If this condition does not hold, one says that the interface is resonant. In Ambroso et al [2,4,5], quite general continuity conditions based on a nonlinear transformation of the unknown were investigated. Following earlier investigations by LeFloch and collaborators [23,31,34,35,36,37] on undercompressive shocks and interfaces, nonconservative hyperbolic systems, and boundary value problems, we stress that additional information coming from physical modeling is necessary in order to single out the relevant continuity conditions (or transmission condition) at the interfaces.…”

“…Following earlier investigations by LeFloch and collaborators [23,31,34,35,36,37] on undercompressive shocks and interfaces, nonconservative hyperbolic systems, and boundary value problems, we stress that additional information coming from physical modeling is necessary in order to single out the relevant continuity conditions (or transmission condition) at the interfaces. Various conditions were introduced and studied in a variety of physical frameworks, ranging from gas dynamics [2] to multiphase flows [1,4].…”

“…Solutions can be shown to exist under general conditions but resonance generally comes at the expense of uniqueness. We refer the reader to [9] for a discussion of scalar equation and to [2] for a distinct behavior exhibited for characteristic but non-resonant interfaces. A selection criterion for discontinuous solutions, therefore, is required.…”

“…[27,28] and [2,4,15]). In these papers, a weak form of the coupling condition (2.3) is formulated in terms of an admissible boundary set, proposed by Dubois and LeFloch [24] and based on the notion of Riemann solutions.…”

Model Adaptation for Hyperbolic Systems with Relaxation

“…The coupling has to be non intrusive because of the complexity of the codes under study, leading to methods which only make use of boundary conditions. This has been the subject of a series of works where several methods of coupling have been proposed for hyperbolic systems of partial differential equations [28,27,5,4,8,16,7,6,25,13]. In all these works, the interface of coupling which separates two different models is fixed.…”