43,137 research outputs found
On the optimization of conservation law models at a junction with inflow and flow distribution controls
The paper proposes a general framework to analyze control problems for
conservation law models on a network. Namely we consider a general class of
junction distribution controls and inflow controls and we establish the
compactness in of a class of flux-traces of solutions. We then derive the
existence of solutions for two optimization problems: (I) the maximization of
an integral functional depending on the flux-traces of solutions evaluated at
points of the incoming and outgoing edges; (II) the minimization of the total
variation of the optimal solutions of problem (I). Finally we provide an
equivalent variational formulation of the min-max problem (II) and we discuss
some numerical simulations for a junction with two incoming and two outgoing
edges.Comment: 29 pages, 14 figure
Well-posedness for a monotone solver for traffic junctions
In this paper we aim at proving well-posedness of solutions obtained as
vanishing viscosity limits for the Cauchy problem on a traffic junction where
incoming and outgoing roads meet. The traffic on each road is governed
by a scalar conservation law , for . Our proof relies upon the complete description of the set
of road-wise constant solutions and its properties, which is of some interest
on its own. Then we introduce a family of Kruzhkov-type adapted entropies at
the junction and state a definition of admissible solution in the same spirit
as in \cite{diehl, ColomboGoatinConstraint, scontrainte, AC_transmission,
germes}
Wave dynamics on networks: method and application to the sine-Gordon equation
We consider a scalar Hamiltonian nonlinear wave equation formulated on
networks; this is a non standard problem because these domains are not locally
homeomorphic to any subset of the Euclidean space. More precisely, we assume
each edge to be a 1D uniform line with end points identified with graph
vertices. The interface conditions at these vertices are introduced and
justified using conservation laws and an homothetic argument. We present a
detailed methodology based on a symplectic finite difference scheme together
with a special treatment at the junctions to solve the problem and apply it to
the sine-Gordon equation. Numerical results on a simple graph containing four
loops show the performance of the scheme for kinks and breathers initial
conditions.Comment: 31 pages, 9 figures, 2 tables, 41 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Failure of Standard Conservation Laws at a Classical Change of Signature
The Divergence Theorem as usually stated cannot be applied across a change of
signature unless it is re-expressed to allow for a finite source term on the
signature change surface. Consequently all conservation laws must also be
`modified', and therefore insistence on conservation of matter across such a
surface cannot be physically justified. The Darmois junction conditions
normally ensure conservation of matter via Israel's identities for the jump in
the energy-momentum density, but not when the signature changes. Modified
identities are derived for this jump when a signature change occurs, and the
resulting surface effects in the conservation laws are calculated. In general,
physical vector fields experience a jump in at least one component, and a
source term may therefore appear in the corresponding conservation law. Thus
current is also not conserved. These surface effects are a consequence of the
change in the character of physical law. The only way to recover standard
conservation laws is to impose restrictions that no realistic cosmological
model can satisfy.Comment: 15pp, figures available on request from Charles Hellaby at
[email protected]
Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions
This paper is concerned with a set of novel coupling conditions for the
one-dimensional Euler system with source terms at a junction of
pipes with possibly different cross-sectional areas. Beside conservation of
mass, we require the equality of the total enthalpy at the junction and that
the specific entropy for pipes with outgoing flow equals the convex combination
of all entropies that belong to pipes with incoming flow. Previously used
coupling conditions include equality of pressure or dynamic pressure. They are
restricted to the special case of a junction having only one pipe with outgoing
flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679--702, 2015]
showed that such pressure-based coupling conditions can produce non-physical
solutions for isothermal flows through the production of mechanical energy. Our
new coupling conditions ensure energy as well as entropy conservation and also
apply to junctions connecting an arbitrary number of pipes with flexible flow
directions. We prove the existence and uniqueness of solutions to the
generalised Riemann problem at a junction in the neighbourhood of constant
stationary states which belong to the subsonic region. This provides the basis
for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for
initial data with sufficiently small total variation.Comment: 17 pages, 2 figure
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