We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux: We refer to such a family as a "germ". It is well known that (CL) admits many different L 1 contractive semigroups, some of which reflect different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the "vanishing viscosity" germ, which is a way of expressing the " -condition" of Diehl ( . We characterize those germs that lead to the L 1 -contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.
This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin in [CG07]. The model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x = 0 (modelling a road light, a toll gate, etc.).We first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function, as developed by Adimurthi et al.[AMG05] and Bürger et al. [BKT09]. We reformulate accordingly the notion of entropy solution introduced in [CG07], and extend the well-posedness results to the L ∞ framework. Then, starting from a general monotone finite volume scheme for the non-constrained conservation law, we produce a simple scheme for the constrained problem and show its convergence. The proof uses a new notion of entropy process solution. Numerical examples modelling a "green wave" are presented.
We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.
International audienceConservation laws of the form $\partial_t u+ \partial_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\mathbf{1}_{x<0}+f_r(\cdot)\mathbf{1}_{x>0}$ were deeply investigated in the last ten years, with a particular emphasis in the case where the fluxes are ''bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of $f_{l,r}$. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications
Abstract. The main goal of this work is to propose a convergent finite volume method for a reaction-diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L 1 compactness argument that mimics the compactness lemma due to S.N. Kruzhkov. The proofs of these results are given in the Appendix.
For the hyperbolic conservation laws with discontinuous flux function there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley-Leverett equations, each notion of solution is uniquely determined by the choice of a "connection", which is the unique stationary solution that takes the form of an undercompressive shock at the interface. To select the appropriate connection, following Kaasschieter (Comput. Geosci., 3(1):23-48, 1999) we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in Cancès (Netw. Heterog. Media, 5(3):635-647, 2010) that the "optimal" connection and the "barrier" connection may appear at the vanishing capillarity limit, we show that the intermediate connections can be relevant and the right notion of solution depends on the physical configuration. In particular, we stress the fact that the "optimal" entropy condition is not always the appropriate one (contrarily to the erroneous interpretation of Kaasschieter's results which is sometimes encountered in the literature). We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley-Leverett equation with interface coupling that retains information from the vanishing capillarity model. We support the theoretical result with numerical examples that illustrate the high efficiency of the algorithm.
In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(·), we are able to model the capacity drop phenomenon at the exit. Existence and stability results for the Cauchy problem with Lipschitz constraint function p(·) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(·). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretization of p(·) may induce non-uniqueness and instability of solutions
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of Simon (Ann. Mat. Pura Appl. 1987). Our approach allows to handle various numerical discretizations both in the space variables and in the time variable. In particular, we can cope quite easily with variable time steps and with multistep time differentiation methods like, e.g., the backward differentiation formula of order 2 (BDF2) scheme. We illustrate our approach by proving the convergence of a two-point flux Finite Volume in space and BDF2 in time approximation of the porous medium equation.
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