Degenerate Convection-diffusion Equations Research Articles

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a roughcoefficient function k(x) . We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV , thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion.

On some upwind difference schemes for the phenomenological sedimentation-consolidation model

In one space dimension, the phenomenological sedimentation-consolidation model reduces to an initial-boundary value problem (IBVP) for a nonlinear strongly degenerate convection-diffusion equation with a non-convex, time-dependent flux function. The frequent assumption that the effective stress of the sediment layer is a function of the local solids concentration only which vanishes below a critical concentration value causes the model to be of mixed hyperbolic-parabolic nature. Consequently, its solutions are discontinuous and entropy solutions must be sought. In this paper, first a (short) guided visit to the mathematical (entropy solution) framework in which the well-posedness of this and a related IBVP can be established is given. This also includes a short discussion of recent existence and uniqueness results for entropy solutions of IBVPs. The entropy solution framework constitutes the point of departure from which numerical methods can be designed and analysed. The main purpose of this paper is to present and demonstrate several finite-difference schemes which can be used to correctly simulate the sedimentation-consolidation model in civil and chemical engineering and in mineral processing applications, i.e., conservative schemes satisfying a discrete entropy principle. Here, finite-difference schemes of upwind type are considered. To some extent, also stability and convergence properties of the proposed schemes are discussed. Performance of the proposed schemes is demonstrated by simulation of two cases of batch settling and one of continuous thickening of flocculated suspensions. The numerical examples focus on a detailed error study, an illustration of the effect of varying the initial datum, and on simulation of practically important thickener operations, respectively.

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

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 Kružkov's method of “doubling the variables.” The application to a sedimentation–consolidation model is illustrated by two numerical examples.

Operator splitting methods for degenerate convection-diffusion equations II: numerical examples with emphasis on reservoir simulation and sedimentation

We present an accurate numerical method for a large class of scalar, strongly degenerate convection–diffusion equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation–consolidation processes. The method is based on splitting the convective and the diffusive terms. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion part is solved by an implicit–explicit finite difference scheme. In addition, one version of the implemented operator splitting method has a mechanism built in for detecting and correcting unphysical entropy loss, which may occur when the time step is large. This mechanism helps us gain a large time step ability for practical computations. A detailed convergence analysis of the operator splitting method was given in Part I. Here we present numerical experiments with the method for examples modelling secondary oil recovery and sedimentation–consolidation processes. We demonstrate that the splitting method resolves sharp gradients accurately, may use large time steps, has first order convergence, exhibits small grid orientation effects, has small mass balance errors, and is rather efficient.

Some results on homogenization of convection-diffusion equations

We consider some models of degenerate convection-diffusion equations with oscillating coefficients. We prove that the homogenization process produces non-local and memory effects when the diffusion is longitudinal. When the diffusion is transverse we obtain a stability result. We also examine parametrized families of diffusion equations involving non-local terms.