Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV , we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat-Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver.

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 consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Kružkov-type notion of entropy solution for this conservation law and prove uniqueness (L 1 stability) of the entropy solution in the BV t class (functions W (x, t) with ∂ t W being a finite measure). The existence of a BV t entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.

The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid-fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.

The one‐dimensional kinematical sedimentation theory for suspensions of small spheres of equal size and density is generalized to polydisperse suspensions and several space dimensions. The resulting mathematical model, obtained by introducing constitutive assumptions and performing a dimensional analysis, is a system of first‐order conservation laws for the concentrations of the solids species coupled to a variant of the Stokes system for incompressible flow describing the mixture. Various flux density vectors for the first‐order system have been proposed in the literature. Some of them cause the first‐order system of conservation laws to be non‐hyperbolic, or to be of mixed hyperbolic‐elliptic type in the bidisperse case. The criterion for ellipticity is equivalent to a well‐known instability criterion predicting phenomena like blobs and viscous fingering in bidisperse sedimentation. We show that loss of hyperbolicity, that is the occurrence of complex eigenvalues of the Jacobian of the first‐order system, can be viewed as an instability criterion for arbitrary polydisperse suspensions, and that for tridisperse mixtures this criterion can be evaluated by a convenient calculation of a discriminant. We determine instability regions (or alternatively prove stability) for three different choices of the flux vector of the first‐order system of conservation laws. Consequently, mixed or non‐hyperbolic, rather than hyperbolic, systems of conservation laws are the appropriate general mathematical framework for polydisperse sedimentation. The stability analysis examines a first‐order system of conservation laws, but its predictions are applicable to the full multidimensional system of model equations. The findings are consistent with experimental evidence and are appropriately embedded into the current state of knowledge of non‐hyperbolic systems of conservation laws.