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Well-posedness in $$BV_t$$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. (English) Zbl 1053.76047
Summary: 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 Kruzhkov-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 $$\partial_tW$$ 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 Engquist-Osher type). A few numerical examples are also presented.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 76T20 Suspensions 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 35R05 PDEs with low regular coefficients and/or low regular data
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