Articlebronze2013

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

Raimund Bürger · Ilja Kröker · Christian Rohde

20 citations0/0 files storedZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

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Probabilistic and Robust Engineering Design Advanced Numerical Methods in Computational Mathematics Model Reduction and Neural Networks

Abstract

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.

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References

1991Stochastic Finite Elements: A Spectral Approach 2003Modeling uncertainty in flow simulations via generalized polynomial chaos 2004Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations 2004Multi-resolution analysis of Wiener-type uncertainty propagation schemes 2008Uncertainty quantification for systems of conservation laws 2012Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data 2010Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems 2008Stochastic finite elements: Computational approaches to stochastic partial differential equations

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2021B-splines on sparse grids for surrogates in uncertainty quantification 2022Arbitrary multi-resolution multi-wavelet-based polynomial chaos expansion for data-driven uncertainty quantification 2016Computational uncertainty quantification for a clarifier-thickener model with several random perturbations: A hybrid stochastic Galerkin approach 2015A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems

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