## Preprint 2012-23

## Raimund Bürger, Ilja Kröker, Christian Rohde:

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

### Abstract:

The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The new 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. Numerical experiments cover one- and two-dimensional situations.

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

**Raimund BüRGER, Ilja KRöKER, Christian ROHDE: ***A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier-thickener unit*. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), vol. 94, 10, pp. 793-817, (2014).