1,242 research outputs found
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}
Time compactness tools for discretized evolution equations and applications to degenerate parabolic PDEs
International audienceWe discuss several techniques for proving compactness of sequences of approximate solutions to discretized evolution PDEs. While the well-known Aubin-Simon kind functional-analytic techniques were recently generalized to the discrete setting by Gallouët and Latché [15], here we discuss direct techniques for estimating the time translates of approximate solutions in the space . One important result is the Kruzhkov time compactness lemma. Further, we describe a specific technique that relies upon the order-preservation property. Motivation comes from studying convergence of finite volume discretizations for various classes of nonlinear degenerate parabolic equations. These and other applications are briefly described
Dissipative interface coupling of conservation laws
International audienceWe give a brief account on the theory of -contractive solvers of the model conservation law with discontinuous flux: \begin{equation*}\label{eq:1D-model} \!\!\leqno(MP)\;\; u_t + (\mathfrak{f}(x,u))_x=0, \quad \mathfrak {f}(x,\cdot)= f^l(\cdot)\char_{x0}, \end{equation*} constructed in the work \cite{AKR-ARMA} of K.H.~Karlsen, N.H.~Risebro and the author. We discuss the modifications that can be used for extending our approach to the multi-dimensional setting and curved flux discontinuity hypersurfaces; the vanishing viscosity case (see \cite{AKR-NHM}) is presented as an illustration. Applications to a road traffic with point constraint and to a coupled particle-fluid interaction model, coming from the joint works \cite{AGS} with P.~Goatin, N.~Seguin and \cite{AS,ALST} with F.~Lagoutiére, N.~Seguin, T.~Takahashi, are presented
Riemann problems with non--local point constraints and capacity drop
In the present note we discuss in details the Riemann problem for a
one--dimensional hyperbolic conservation law subject to a point constraint. We
investigate how the regularity of the constraint operator impacts the
well--posedness of the problem, namely in the case, relevant for numerical
applications, of a discretized exit capacity. We devote particular attention to
the case in which the constraint is given by a non--local operator depending on
the solution itself. We provide several explicit examples. We also give the
detailed proof of some results announced in the paper [Andreainov, Donadello,
Rosini, "Crowd dynamics and conservation laws with non--local point constraints
and capacity drop", which is devoted to existence and stability for a more
general class of Cauchy problems subject to Lipschitz continuous non--local
point constraints.Comment: 19 pages, 6 figures. arXiv admin note: substantial text overlap with
arXiv:1304.628
Convergence of discrete duality finite volume schemes for the cardiac bidomain model
We prove convergence of discrete duality finite volume (DDFV) schemes on
distorted meshes for a class of simplified macroscopic bidomain models of the
electrical activity in the heart. Both time-implicit and linearised
time-implicit schemes are treated. A short description is given of the 3D DDFV
meshes and of some of the associated discrete calculus tools. Several numerical
tests are presented
Well-posedness results for triply nonlinear degenerate parabolic equations
We study the well-posedness of triply nonlinear degenerate
elliptic-parabolic-hyperbolic problem in a bounded domain with
homogeneous Dirichlet boundary conditions. The nonlinearities and
are supposed to be continuous non-decreasing, and the nonlinearity
falls within the Leray-Lions framework. Some restrictions
are imposed on the dependence of on
and also on the set where degenerates. A model case is
with which is strictly increasing except on a locally finite number of
segments, and which is of the Leray-Lions kind. We are
interested in existence, uniqueness and stability of entropy solutions. If
, we obtain a general continuous dependence result on data
and nonlinearities . Similar result
is shown for the degenerate elliptic problem which corresponds to the case of
and general non-decreasing surjective . Existence, uniqueness
and continuous dependence on data are shown when and
is continuous
NEW APPROACHES TO DESCRIBING ADMISSIBILITY OF SOLUTIONS OF SCALAR CONSERVATION LAWS WITH DISCONTINUOUS FLUX
International audienceHyperbolic conservation laws of the form u_t + div f(t, x; u) = 0 with discontinuous in (t, x) flux function f attracted much attention in last 20 years, because of the difficulties of adaptation of the classical Kruzhkov approach developed for the smooth case. In the discontinuous-flux case, non-uniqueness of mathematically consistent admissibility criteria results in infinitely many different notions of solution. A way to describe all the resulting L1 -contractive solvers within a unified approach was proposed in the work [Andreianov, Karlsen, Risebro, 2011]. We briefly recall the ideas and re-sults developed there for the model one-dimensional case with f(t, x; u) = f_l (u)1_{x0} and highlight the main hints needed to address the multi-dimensional situation with curved interfaces. Then we discuss two recent developments in the subject which permit to better understand the issue of admissibility of solutions in relation with specific modeling assumptions; they also bring useful numerical approximation strategies. A new characterization of limits of vanishing viscosity approxi-mation proposed in [Andreianov and Mitrovic, 2014] permits to encode admissibility in singular but intuitively appealing entropy inequalities. Transmission maps introduced in ([Andreianov andCan es, 2014]) have applications in modeling flows in strongly heterogeneous porous media and lead to a simple algorithm for numerical approximation of the associated solutions. Moreover, in order to embed all the aforementioned results into a natural framework, we put forward the concept of interface coupling conditions (ICC) which role is analogous to the role of boundary conditions for boundary-value problems. We link this concept to known examples and techniques
A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rocks media
International audienceWe discuss the convergence of the upstream phase-by-phase scheme (or upstream mobility scheme) towards the vanishing capillarity solution for immiscible incompressible two-phase flows in porous media made of several rock types. Troubles in the convergence where recently pointed out in [S. Mishra & J. Jaffré, Comput. Geosci., 2010] and [S. Tveit & I. Aavatsmark, Comput. Geosci, 2012]. In this paper, we clarify the notion of vanishing capillarity solution, stressing the fact that the physically relevant notion of solution differs from the one inferred from the results of [E. F. Kaasschieter, Comput. Geosci., 1999]. In particular, we point out that the vanishing capillarity solution de- pends on the formally neglected capillary pressure curves, as it was recently proven in by the authors [B. Andreianov & C. Canc'es, Comput. Geosci., 2013]. Then, we propose a numerical procedure based on the hybridization of the interfaces that converges towards the vanishing capillarity solution. Numerical illustrations are provided
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