# Preprint 2012-020

# Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

## R. Bürger and S. Diehl

**Abstract:**
The sedimentation of a suspension of small particles
dispersed in a viscous fluid
can be described by a scalar, nonlinear conservation law,
whose flux function usually has one inflection point.
The identification of the flux function
is a problem of theoretical interest and practical importance
for the implementation of plant-scale
simulators of continuous sedimentation.
For a real suspension, the Kynch test and the Diehl test,
which are based on an initially homogenous suspension
either filling the whole settling column
or being initially located above clear liquid, respectively,
provide data points that represent curved
(convex or concave, respectively)
suspension-supernate interfaces
from which it is possible to reconstruct portions
of the flux function to either side of the inflection point.
Several functional forms can be employed
to generate a provably convex or concave,
twice differentiable accurate approximation
of these data points via the solution
of a constrained least-squares minimization problem.
The resulting spline-like estimated trajectory
can be converted into an explicit formula for the flux function.
It is proved that the inverse problem of flux identification
solved this way has a unique solution.
The problem of gluing together the portions of the flux function
from the Kynch and Diehl tests is addressed.
Examples involving synthetic data are presented.