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.