We show how existingmodels for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions formingcompressible sediments (“sedimentation with compression” or “sedimentation-consolidation process”). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov–Tadmor central difference scheme for convection-diffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.

The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation-consolidation processes of flocculated suspensions in clarifierthickener units. This model appears in two variants for cylindrical and variable cross-sectional area units, respectively (Models 1 and 2). In both cases, the governing equation is a scalar, strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable. The initial value problem for this equation is analyzed for Model 1. We introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition. A limited analysis of steady states as desired stationary modes of operation is performed. Numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier-thickeners.

We describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convection-diffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The conservation law is solved by a front tracking method, while the diffusion equation is here solved by a finite difference scheme. The numerical methods are unconditionally stable in the sense that the (splitting) time step is not restricted by the spatial discretization parameter. The strategy is designed to handle all combinations of convection and diffusion (including the purely hyperbolic case). Two numerical examples are presented to highlight the features of the methods, and the potential for parallel implementation is discussed.

. We study entropy solutions of nonlinear degenerate parabolic equations of form u t + div k(x)f(u) = A(u), where k(x) is a vector-valued function and f(u); A(u) are scalar functions. We prove a result concerning the continuous dependence on the initial data, the ux function k(x)f(u), and the diusion function A(u). This paper complements previous work [7] by two of the authors, which contained a continuous dependence result concerning the initial data and the ux function k(x)f(u). 1. Introduction In this paper we are concerned with entropy solutions of the initial value problem (1.1) u t + div k(x)f(u) = A(u); u(x; 0) = u 0 (x) for (x; t) 2 T = R d (0; T ) with T > xed. In (1.1), u(x; t) is the scalar unknown function that is sought, k(x)f(u) is the ux function, and A = A(u) is the diusion function. We always assume that k : R d ! R, f : R ! R, and A : R ! R satisfy (1.2) ( k 2 W 1;1 loc (R d ); k; divk 2 L 1 (R d ); f 2 Lip loc (R); f(0) = 0; A 2 Li...

. Following the lead of Carrillo [6], recently several authors have used Kruzkov's device of \doubling the variables" to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order dierential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result found in Karlsen and Risebro [14] to a class of degenerate parabolic equations with spatially dependent second order dierential operator. The class is large enough to encompass several interesting nonlinear partial dierential equations coming from the theory of porous media ow and the phenomenological theory of sedimentation-consolidation processes. 1.

We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation # t u + #x x A(u), A # () where the coefficient #(x) is possibly discontinuous and f(u) is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as # 0 in a suitable sequence of smooth approximations solving the problem above with the transport flux #(x)f() replaced by ## (x)f() and the diffusion function A() replaced by A# (), where ## () is smooth and A # # () > 0. The main technical challenge is to deal with the fact that the total variation |u# | uniformly in #, and hence one cannot derive directly strong convergence of . In the purely hyperbolic case (A # 0), where existence has already been established by a number of authors, all existence results to date have used a singular mapping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy) estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term.

. We analyse implicit monotone finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion equations in one spatial dimension. Since we allow strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. We thus choose to work with weak solutions that belong to the BV class and, in addition, satisfy an entropy condition. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. This paper complements our previous work [8] on explicit monotone schemes. 1. Degenerate Convection-Diffusion Equations We are interested in finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion problems of the form @ t u + @ x f(u) = @ x (k(u)@ x u); u(x; 0) = u 0 (x); (1) where (x; t) 2 QT = R\Theta (0; T ) and u 0 ; f; k are given, sufficiently smooth functions. For later use, we need a conservative-form version of (1), @ t u + @ x f(u) = @ 2 x K(u)...

We propose a new notion of weak solutions (dissipative solutions) for nonisotropic, degenerate, second order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations

The paper focuses on the uniqueness issue for scalar convection-diffusion equations where both the convective flux and diffusion functions have a spatial discontinuity. An interface entropy condition is proposed at such a spatial discontinuity. It implies the Kruˇzkov-type entropy condition presented by Karlsen et al. [Trans. Royal Norwegian Society Sci. Letters 3, 49 pp, 2003]. They proved uniqueness when the convective flux function satisfies an additional ‘crossing condition’. The crossing condition becomes redundant with the entropy condition proposed here. Thereby, more general flux functions are allowed. Another advantage of the entropy condition is its simple geometrical interpretation, which facilitates the construction of stationary solutions.

Abstract. We develop a general L1 –framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply our L1 –framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an L1 error estimate for an upwind-central finite difference scheme.