Raimund Bürger, Julio Careaga, Stefan Diehl:
Flux identification for scalar conservation laws modelling sedimentation in vessels with varying cross-sectional area
A method is presented for the identification of a non-convex flux function of a hyperbolic scalar conservation law that models sedimentation of solid particles in a liquid. While all previous identification methods are based on data obtained from settling tests in cylindrical vessels, the novel approach is based on the richer solution behaviour produced in a vessel with downward-decreasing cross-sectional area. Except for the initial homogeneous concentration, the data given for the present inverse problem are the location of the decline of the supernatant-suspension interface as a function of time. The inverse problem is solved by utilizing the construction of solutions of the direct problem by the method of characteristics. In theory, the entire flux function can be estimated from only one batch settling experiment, and the solution is given by parametric and explicit formulas for the flux function. The method is tested on synthetic data (for example, generated by numerical simulations with a known flux) and on published experimental data.