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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.
Reviewer: Reviewer (Berlin)

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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