Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations

Abstract

We consider consistent, conservative-form, monotone nite di erence schemes for nonlinear convection-di usion equations in one space dimension. Since we allow the di usion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin 30] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase ow equation and hyperbolic conservation laws. The di erence schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone di erence approximations of conservation laws, the main di culty in obtaining a similar convergence theory in the present context is to show that the approximations are L 1 Lipschitz continuous in the time variable (this is trivial for conservation laws). This continuity result is in turn intimately related to the regularity properties possessed by the (strongly degenerate) discrete di usion term. We provide the necessary regularity estimates on the di usion term by deriving and carefully analysing a linear di erence equation satis ed by the numerical ux of the di erence schemes. x1. Introduction. We are interested in monotone nite di erence approximations of nonlinear, possibly strongly degenerate, convection-di usion problems of form (1) (@ t u + @ x f(u) = @ x (k(u)@ x u); (x; t) 2 Q T R h0; Ti ; k(u) 0; u(x; 0) = u 0 (x); where the initial condition u 0 (x), the convection ux f(u) and the di usion ux k(u) 0 are given, su ciently regular functions. Convection-di usion equations arise in a variety of applications, among others turbulence, tra c ow, nancial modelling, front propagation, two phase ow in oil reservoirs, and in models describing certain sedimentation processes. When (1) is non-degenerate, i.e., k(u) > 0, it is well known that (1) admits a unique classical solution 21]. This contrasts with the degenerate case where k(u) may vanish for some values of u. A simple example of a degenerate equation is the porous medium equation, (2) @ t u = @ x (u m) ; m > 1; 1991 Mathematics Subject Classi cation. 65M12, 35K65, 35L65. Key words and phrases. Degenerate convection-di usion equations, BV solutions, entropy condition, monotone nite di erence schemes, convergence. Karlsen has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…Discretization of the aforementioned hyperbolic, porous medium, convection-diffusion, and ellipticparabolic equations by finite volume methods is quite standard by now and often used in engeneering practice. We refer to [48,31,3,44,45,61,57,68,79,49,67,12,10,11,42] and references therein for different convergence results and numerical experiments. For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8.…”

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

“…Finally we present the following test: The previous equation is the adaption of a similar test for convection-diffusion equations which can be found in [20]. In our analysis, we did not address the possibility for a strongly degenerate equation to present more than a weak solution, among which one physical solution has to be selected by introducing an entropy function, as in the case of hyperbolic conservation laws.…”

“…In our analysis, we did not address the possibility for a strongly degenerate equation to present more than a weak solution, among which one physical solution has to be selected by introducing an entropy function, as in the case of hyperbolic conservation laws. For a detailed presentation of the entropy issue for the parabolic equations, see [20] and references therein.…”

On uniqueness techniques for degenerate convection-diffusion problems

“…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. …”