Discrete Duality Finite Volume Schemes for Doubly Nonlinear Degenerate Hyperbolic-Parabolic Equations

Abstract

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.

The Semigroup Approach to Conservation Laws with Discontinuous Flux

“…This is a consequence of the Kruzhkov inequalities in domains Ω l,r ; see, e.g., [26,4,18] and references therein. In the sequel, we will always select the time-continuous representative of u; in particular, the initial condition can be taken in the sense u(0, ·) = u 0 .…”

On uniqueness techniques for degenerate convection-diffusion problems

“…Namely, the test function ξ in (16) is allowed to be nonzero at the boundary only for k ≥ 0 (in the "sign + " inequalities) or for k ≤ 0 (in the "sign − " inequalities). In the doubling of variables procedure, the positive and negative parts of the two solutions are separated and treated apart, using entropy inequalities (16) for the aforementioned couples (k, ξ) (see, e.g., [7,Lemma A.2] for the elementary calculation underlying this separation). The argument is lengthy; we refer to the original paper [26] and to [7,Lemma A.5] where the different steps of the proof "near the boundary" are highlighted.…”

“…Let us mention that this technique of [26] for the homogeneous problem works for the convection-diffusion fluxes (2) under the assumptions of Proposition 3.6. One can follow, e.g., the arguments of [7,Lemma A.5] with the calculations of Proposition 3.6 in hand, in order to treat the neighbourhood of the boundary.…”

“…In [26], the appropriate notion of entropy solution was established, and this definition (or, sometimes, parts of the uniqueness techniques of [26]) led to many developments; among them, let us mention [2,3,5,4,6,8,9,10,11,12,13,18,19,24,25,27,29,30,34,37,38,39,41,42,45,46,47,48,52,53,57,58,59]. Also numerical aspects of the problem were investigated; see, e.g., [7,32,33,35,40,49].The notion of entropy solution (or, as in the present paper, entropy solutions techniques used on weak solutions) was retained by most of the authors; yet, let us mention the version of Bendahmane and Karlsen [18,19] We thank Safimba Soma for fruitful discussions on the subject of this note, and the anonymous referee for a careful reading and very pertinent remarks. …”

“…Yet the Stefan type problems with continuous F serve as a playground for the wide class of practically important hyperbolic-parabolic-elliptic problems (see in particular [26,47,48,59,7]); for these problems, entropy inequalities and the doubling of variables remain the essential technique. It is an open question how to transfer to this context the techniques of [9,12] or those of [11] recalled in this paper; some work in this direction is in progress.…”