Preprint 2003-006

Well-Posedness in BVt and Convergence of a Difference Scheme for Continuous Sedimentation in Ideal Clarifier-Thickener Units

R. Bürger, K. H. Karlsen, N. H. Risebro, and J. D. Towers

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 Kru\v{z}kov-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 $\pt W$ 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.

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R. Bürger, <>
K. H. Karlsen, <>
N. H. Risebro, <>
J. D. Towers, <>
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<> January 17 2003.

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