On Strongly Degenerate Convection–Diffusion Problems Modeling Sedimentation–Consolidation Processes

Abstract

We investigate initial-boundary value problems for a quasilinear strongly degenerate convection᎐diffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentation᎐ consolidation processes. The existence of entropy solutions belonging to BV is shown by the vanishing viscosity method. The existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient. New uniqueness proofs for entropy solutions are also presented. These proofs rely on a recent extension to second-order equations of Kruzkov's method of ''doubling thě variables.'' The application to a sedimentation᎐consolidation model is illustrated by two numerical examples. ᮊ

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

“…Similarly, ν K * = ν K * | S is the one of the vectors ν * S , −ν * S that is exterior to K * . In fact, formulas (24), (25) can be conveniently expressed in terms of vector products involving the discrete field F S and specific geometric objects depicted in Figure 3 (see [8,7]). …”

“…Here S n K , S n L are given by (30), (32); g K,L , g K * ,L * are some numerical convection fuxes satisfying (28); ν K , ν K * for S given have the same meaning as in (24), (25); finally, …”

“…Since then, many authors extended the Carrillo results in various directions (see e.g. [30,59,66,73,25,46,60,62,67,49,5,13]). Some additional techniques are required for anisotropic diffusion problems, where a kinetic approach (see Chen, Perthame [35]) and an accurate entropic approach (see Bendahmane, Karlsen [17,18]) were developed in the few last years; see also Souganidis, Perthame [75] and and Chen, Karlsen [34].…”

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

“…The initial-boundary value problem for (1.1) that models batch sedimentation is analyzed in [8], where the existence of BV entropy solutions is shown via the vanishing viscosity method, while their uniqueness follows by a technique introduced by Carrillo [15]. On the other hand, Evje and Karlsen [25] show that explicit monotone finite difference schemes, which were introduced by Harten et al [32] and Crandall and Majda [18] for conservation laws, converge to BV entropy solutions for initial-value problems of strongly degenerate parabolic equations.…”

On uniqueness techniques for degenerate convection-diffusion problems

“…Study of degenerate parabolic problems has undergone a considerable progress in the last ten years, thanks to the fundamental paper of J. Carrillo [26] in which the Kruzhkov device of doubling of variables was extended to hyperbolic-parabolic-elliptic problems of the form j(v)−div(f (v)+∇ϕ(v)) = 0, and a technique for treating the homogeneous Dirichlet boundary conditions was put forward. 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].…”