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In the applications of this model, which include mineral processing and wastewater 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. The presented hybrid stochastic Galerkin method is devised so as to extend the polynomial chaos approximation by multiresolution discretization in the stochastic space. This approach leads to a deterministic hyperbolic system, which is partially decoupled and therefore suitable for efficient parallelisation. Stochastic adaptivity reduces the computational effort. Several numerical experiments are presented.", "issn" : "0098-1354", "doi" : "http://dx.doi.org/10.1016/j.compchemeng.2016.02.016", "bibtexKey": "Barth201611" } , { "type" : "Publication", "id" : "https://puma.ub.uni-stuttgart.de/bibtex/2d243188d800231a5e87752565bf2f54c/ik", "tags" : [ "HSG","Stochastic-Galerkin","UQ","clarifier-thickener","hyperbolic","myown" ], "intraHash" : "d243188d800231a5e87752565bf2f54c", "interHash" : "3e75df22fa8ec60ad20e3c0a1e9f2a72", "label" : "A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier-thickener unit", "user" : "ik", "description" : "A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier-thickener unit - Bürger - 2013 - ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik - Wiley Online Library", "date" : "2016-04-26 10:37:26", "changeDate" : "2016-04-26 12:08:02", "count" : 1, "pub-type": "article", "journal": "ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik","publisher":"WILEY-VCH Verlag", "year": "2014", "url": "http://dx.doi.org/10.1002/zamm.201200174", "author": [ "R. Bürger","I. Kröker","C. Rohde" ], "authors": [ {"first" : "R.", "last" : "Bürger"}, {"first" : "I.", "last" : "Kröker"}, {"first" : "C.", "last" : "Rohde"} ], "volume": "94","number": "10","pages": "793--817","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.", "issn" : "1521-4001", "doi" : "10.1002/zamm.201200174", "bibtexKey": "ZAMM:ZAMM201200174" } , { "type" : "Publication", "id" : "https://puma.ub.uni-stuttgart.de/bibtex/2fbdb6c6af9777516523af2681bbede3a/ik", "tags" : [ "hyperbolic","multiplicative-noise","myown" ], "intraHash" : "fbdb6c6af9777516523af2681bbede3a", "interHash" : "64860e369eaa4c497300fa7d8ca256c6", "label" : "Finite volume schemes for hyperbolic balance laws with multiplicative noise", "user" : "ik", "description" : "", "date" : "2016-04-26 10:34:30", "changeDate" : "2016-04-26 08:35:20", "count" : 6, "pub-type": "article", "journal": "Applied Numerical Mathematics", "year": "2012", "url": "http://www.sciencedirect.com/science/article/pii/S016892741100033X", "author": [ "I. Kröker","C. Rohde" ], "authors": [ {"first" : "I.", "last" : "Kröker"}, {"first" : "C.", "last" : "Rohde"} ], "volume": "62","number": "4","pages": "441 - 456","note": "Third Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2010)","abstract": "We consider finite volume schemes for a scalar stochastic balance law with multiplicative noise. For a class of monotone numerical fluxes we establish the pathwise convergence of a semi-discrete finite volume solution towards a stochastic entropy solution. Main tool is a stochastic version of the compensated compactness approach. The approach relies solely on L p -estimates. It avoids the use of a maximum principle and total-variation estimates. These are typical tools in the deterministic case but are not available for the non-deterministic model. Numerical results illustrate the analytical findings.", "issn" : "0168-9274", "doi" : "http://dx.doi.org/10.1016/j.apnum.2011.01.011", "bibtexKey": "Kröker2012441" } ] }