Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Abstract

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 of theoretical interest and practical importance for 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.