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 $L^1$. 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 $L^1$-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

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

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 $m$ incoming and $n$ outgoing roads meet. The traffic on each road is governed by a scalar conservation law $\rho_{h,t} + f_h(\rho_h)_x = 0$, for $h\in \{1,\ldots, m+n\}$. 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}

Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition

International audienceWe study the Cauchy-Dirichlet problem for the elliptic-parabolic equation $$b(u)_t +\div F(u) - \Delta u=f$$ in a bounded domain. We do not assume the structure condition ''$b(z)=b(\hat z) \Rightarrow F(z)=F(\hat z)$''. Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold \cite{AmmarWittbold} where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux $F$, we justify the uniqueness of $u$ (the uniqueness of $b(u)$ is well-known) and prove the continuous dependence in $L^1$ for the case of strongly convergent finite energy data. We also prove convergence of the $\varepsilon$-discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux $F$, we show that the problem admits a maximal and a minimal solution

Well-posedness of general boundary-value problems for scalar conservation laws

International audienceIn this paper we investigate well-posedness for the problem u_t+ \div \ph(u)=f on (0,T)\!\times\!\Om, \Om \subset \R^N, with initial condition $u(0,\cdot)=u_0$ on \Om and with general dissipative boundary conditions $\varphi(u)\cdot \nu \in \beta_{(t,x)}(u)$ on (0,T)\!\times\!\ptl\Om. Here for a.e. (t,x)\in(0,T)\!\times\!\ptl\Om, $\beta_{(t,x)}(\cdot)$ is a maximal monotone graph on $\R$. This includes, as particular cases, Dirichlet, Neumann, Robin, obstacle boundary conditions and their piecewise combinations. As for the well-studied case of the Dirichlet condition, one has to interprete the {\it formal boundary condition} given by $\beta$ by replacing it with the adequate {\it effective boundary condition}. Such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation. We claim that the formal boundary condition given by $\beta$ should be interpreted as the effective boundary condition given by another monotone graph $\tilde \beta$, which is defined from $\beta$ by the projection procedure we describe. We give several equivalent definitions of entropy solutions associated with $\tilde \beta$ (and thus also with $\beta$). For the notion of solution defined in this way, we prove existence, uniqueness and $L^1$ contraction, monotone and continuous dependence on the graph $\beta$. Convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results

The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions

6 pagesInternational audienceWe consider hyperbolic scalar conservation laws with discontinuous flux function of the type \partial_t u + \partial_x f(x,u) = 0 \text{\;\;with\;\;} f(x,u) = f_L(u) \Char_{\R^-}(x) + f_R(u) \Char_{\R^+}(x). Here $f_{L,R}$ are compatible bell-shaped flux functions as appear in numerous applications. It was shown by Adimurthi, S. Mishra, G. D. V. Gowda ({\it J. Hyperbolic Differ. Equ. 2 (4) (2005) 783-837)} and R. Bürger, K. H. Karlsen and J. D. Towers ({\it SIAM J. Numer. Anal. 47~(3) (2009) 1684--1712}) that several notions of solution make sense, according to a choice of the so-called $(A,B)$-connection. In this note, we remark that every choice of connection $(A,B)$ corresponds to a limitation of the flux under the form $f(u)|_{x=0}\leq \bar F$, first introduced by R. M. Colombo and P. Goatin ({\it J. Differential Equations 234 (2) (2007) 654-675}). Hence we derive a very simple and cheap to compute explicit formula for the Godunov numerical flux across the interface $\{x=0\}$, for each choice of connection. This gives a simple-to-use numerical scheme governed only by the parameter $\bar F$. A numerical illustration is provided