On the coupling of systems of hyperbolic conservation laws with ordinary differential equations

Abstract

Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided by the boundary condition. For each of the examples considered, numerical integrations are provided.

ADER schemes and high order coupling on networks of hyperbolic conservation laws

“…Following the results of [26,9], the number of coupling conditions c has to coincide with the number of characteristics running out of the vertex. In order to maintain a fixed number of coupling conditions over time, we require for each edge i the eigenvalues λ i j , j = 1, .…”

“…Networks of hyperbolic conservation laws occur in many applications such as the human circulatory system [1,2,3,4], gas pipelines [5,6,7], water [8,9,10,11] and road networks [12,13]. For all these applications accurate and stable numerical methods are needed.…”

“…Special attention has to be given to the coupling conditions. The direct solving of the coupling conditions only provides first order information, which can either be used directly by applying a Godunov scheme [6,23] or to fill corresponding ghost cells [9,24] at the boundary. A second order approach is studied in [25].…”

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

“…The study of coupled PDE-ODE systems is not new in the conservation laws framework, we refer the reader to [3,7,10,18]. Nevertheless, the problem posed here is slightly different.…”

“…On one side, we deal with a strong coupling with the PDE and the ODE affecting each other, unlike [7,10], where the PDE solution does not depend on the ODE. On the other side, even if the ODE has discontinuous right-hand side, the particular definition of the model allows us to consider classical Carathéodory solutions, as in [3,2,4,7], instead of the weaker Filippov's generalized solutions needed in [10,18].…”