Well-posedness in BV t 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 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.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

“…Kružkov's theory does not apply when γ is discontinuous. In [19], a variant of Kružkov's notion of entropy weak solution for (1.3) that accounts for the discontinuities in γ is introduced and existence and uniqueness (stability) of such entropy solutions in a certain functional class are proved. The existence of such solutions follows from the convergence of various numerical schemes such as front tracking [50], a relaxation scheme [51,52], and upwind difference schemes [19].…”

“…Note that our pointwise discretization of γ, (3.2), follows the usage of [16,20,21,22], but differs from that of [19], where γ is discretized by cell averages taken over the cells [x j , x j+1 ), where x j := j∆x, j ∈ Z. The important point is that in both cases, the discretization of γ is staggered with respect to that of the conserved quantity u, and this property greatly facilitates the convergence analysis of the numerical schemes.…”

“…This equation is discretized in space by a first-order conservative finite volume scheme using the Engquist-Osher approximation, for which convergence results for our class of problems are available [16,19,20,21,22]. For time discretization an explicit Euler scheme is used.…”

Operating Charts for Continuous Sedimentation II: Step Responses

“…In [26], existence and uniqueness of solutions were established only locally in time in the class of piecewise differentiable functions and with some further assumptions on piecewise monotonicity. It is only lately that global existence and uniqueness have been established by Bürger et al [31,32] and Karlsen and Towers [33]. In [31], the front-tracking method was utilized and approximate solutions were constructed as in [26].…”

“…Global existence was then established by proving convergence of the numerical front-tracking method. In [32], a well-posed entropy solution framework was established and uniqueness was shown. Existence of an entropy solution was proved with the aim of an upwind finite difference scheme of Engquist-Osher type.…”