Ilja Kröker

Computational uncertainty quantification for a clarifier-thickener model with several random perturbations: A hybrid stochastic Galerkin approach

Abstract Continuous sedimentation processes in a clarifier-thickener unit can be described by a scalar nonlinear conservation law whose flux density function is discontinuous with respect to the spatial position. 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.

Stochastic Modeling for Heterogeneous Two-Phase Flow

The simulation of multiphase flow problems in porous media often requires techniques for uncertainty quantification to represent parameter values that are not known exactly. The use of the stochastic Galerkin approach becomes very complex in view of the highly nonlinear flow equations. On the other hand collocation-like methods suffer from low convergence rates. To overcome these difficulties we present a hybrid stochastic Galerkin finite volume method (HSG-FV) that is in particular well-suited for parallel computations. The new approach is applied to specific two-phase flow problems including the example of a porous medium with a spatially random change in mobility. We emphasize in particular the issue of parallel scalability of the overall method.

A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems

Simulating in flow problems in porous media often requires techniques for uncertainty quantification in order to represent parameter values that are not given exactly. Straightforward Monte-Carlo (MC) methods have a limited efficiency due to slow convergence. Better convergence in low-parametric problems can be achieved with polynomial chaos expansion (PCE) techniques. The PCE approach yields a coupled deterministic system to be solved. The degree of coupling increases with the non-linearity of the considered equations and with the order of polynomial expansion. This fact increases the computational effort of PCE and significantly reduces the scalability in parallelization. We present an application of the hybrid stochastic Galerkin finite volume (HSG-FV) method to a two-phase flow problem in two space dimensions. The method extends the classical polynomial chaos expansion by a multi-element discretization in the probability space of the parameters. It leads to a deterministic system that is coupled to a lesser degree than in element-free PCE versions, respectively, fully decoupled in stochastic elements (SE). Therefore, the HSG-FV method allows for more efficient parallelization. For the further reduction of complexity, we present a new stochastic adaptivity method. We present numerical examples in two spatial dimensions with linear and nonlinear fractional flow functions in the two-phase flow problem. The flow functions might depend in a discontinuous manner on the unknown spatial position of porous-medium heterogeneities. Finally, we discuss the interplay of the new method with spatial adaptivity per SE for these problems.

Uncertainty Quantification for a Clarifier–Thickener Model with Random Feed

The continuous sedimentation process in a clarifier–thickener can be described by a scalar nonlinear conservation law for the solid volume fraction. The flux is discontinuous with respect to space due to the feed mechanism. Typically the feed flux cannot be given in an exact manner. To quantify uncertainty the unknown solid concentration and the feed bulk flow are expressed by polynomial chaos. A deterministic hyperbolic system for a finite number of stochastic moments is constructed. For the resulting high-dimensional system a simple finite volume scheme is presented. Numerical experiments cover one- and two-dimensional situations.

Computational uncertainty quantification for some strongly degenerate parabolic convection–diffusion equations

Abstract Strongly degenerate parabolic convection–diffusion equations arise as governing equations in a number of applications such as traffic flow with driver reaction and anticipation distance and sedimentation of solid–liquid suspensions in mineral processing and wastewater treatment. In these applications several parameters that define the convective flux function and the degenerating diffusion coefficient are subject to stochastic variability. A method to evaluate the variability of the solution of the governing partial differential equation in response to that of the parameters is presented. To this end, a generalized polynomial chaos (gPC) expansion of the solution is approximated by its projection onto a finite-dimensional space of piecewise polynomial functions defined on a suitable discretization of the stochastic domain, according to the basic principle of the hybrid stochastic Galerkin (HSG) approach. This approach is combined with a finite volume (FV) method, resulting in a so-called FV–HSG method, to compute the sought deterministic coefficient functions of the truncated polynomial-chaos-based expansion of the solution. Since the stochastic parameter space is now spanned by piecewise polynomial functions, one may employ the numerical result to compute the reconstruction of the numerical solution for arbitrary values of the random variables. The expectation, the variance or other stochastic quantities of the solution (as functions of time and position) can also be computed from these coefficient functions. The method is illustrated by a number of numerical examples.

Hybrid Stochastic Galerkin Finite Volumes for the Diffusively Corrected Lighthill-Whitham-Richards Traffic Model

The vehicular traffic on an infinite or circular highway can be represented by the diffusively corrected Lighthill-Whitham-Richards (DCLWR) model, but the uncertain knowledge requires to deal with uncertain model parameters. The stochastic Galerkin based methods allow to transform a randomly perturbed PDE to a high-dimensional deterministic system, where the dimension of the system increases rapidly with the increasing number of the uncertain parameters. In this work we consider the application of the hybrid stochastic Galerkin (HSG) finite volume method to the uncertain DCLWR model, which is represented by a random perturbed strongly degenerate parabolic equation. We present the resulting high–dimensional system and corresponding finite volume method. Numerical examples cover the scalar problem with four uncertain parameters.

Intrusive uncertainty quantification for hyperbolic-elliptic systems governing two-phase flow in heterogeneous porous media

We model random locations of spatial interfaces of heterogeneities in porous media by means of the hybrid stochastic Galerkin (HSG) approach. This approach extends the concept of the generalized polynomial chaos (PC) expansion for a multi-element decomposition of the multidimensional stochastic space. In this way, the physically two-dimensional hyperbolic-elliptic fractional flow formulation for two-phase flow in heterogeneous porous media is transformed from a random partial differential equation into a deterministic system for the coefficients of the PC expansion of the primary unknown saturation, total velocity, and global pressure. The hyperbolic part is discretized based on a central-upwind finite volume scheme along with a mixed finite element method for the elliptic part. The latter partly uses the tensor product structure of the saddle point system together with appropriate preconditioning to speed up the computations. Since we use the sequential implicit pressure explicit saturation (IMPES) approach, the elliptic part is solved in every time step. The proposed method is particularly well-suited for parallel computations and allows for the consideration of a huge variety of complex flow problems. We illustrate the power of the method by means of several striking numerical examples of different complexities including their numerical convergence analysis.

Finite Volume Methods for Hyperbolic Partial Differential Equations with Spatial Noise

Various real-world applications require the consideration of the influence of uncertain parameters on the solution to some nonlinear hyperbolic problem. The topic of this paper is to study the solution to a nonlinear hyperbolic PDE perturbed by a spatial noise term. In the first part of this paper, a definition of the corresponding stochastic entropy solution is defined and the required properties for the existence and uniqueness of the defined solution are discussed. The second part is devoted to numerical simulation. An approximation of the spatial noise is introduced. Further, the influence of the noise on the solution to the nonlinear hyperbolic problems is investigated. Several nonlinear numerical examples illustrate the discussion.