## Preprint 2015-34

## Andrea Barth, Raimund Bürger, Ilja Kröker, Christian Rohde:

### A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier-thickener unit with several ramdom sources

### Abstract:

The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction whose flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. In the applications of this model, which include mineral processing and wastwater treatment, the rate and composition of the feed flow cannot be given deterministically. Efficient numerical simulation is required to quantify the effect of uncertainty in these control parameters in terms of the response of the clarifier-thickener system. Thus, the problem at hand is one of uncertainty quantification for nonlinear hyperbolic problems with several random perturbations. To solve it, a hybrid stochastic Galerkin (HSG) method is devised that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The approach leads to a deterministic hyperbolic system for a finite number of stochastic moments which is however partially decoupled and thus allows efficient parallelisation. The complexity of the problem is further reduced by stochastic adaptivity. For the approximate solution of the resulting high-dimensional system a finite volume scheme is introduced. Several numerical experiments are presented.

This preprint gave rise to the following definitive publication(s):

**Andrea BARTH, Raimund BüRGER, Ilja KRöKER, Christian ROHDE: ***Computational uncertainty quantification for a clarifier-thickener model with several random perturbations: A hybrid stochastic Galerkin approach*. Computers & Chemical Engineering, vol. 89, pp. 11-26, (2016).