WELL-POSEDNESS IN BVt AND CONVERGENCE OF A DIFFERENCE SCHEME FOR CONTINUOUS SEDIMENTATION IN IDEAL CLARIFIER-THICKENER UNITS

Abstract

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 Kruzkov-type notion of entropy solution for this conservation law and prove iniqueness ($L_1$ stability) of the entropy solution in the $BV_t$ class (functions $W(x,t)$ with $\partial_tW$ 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.