Stefan Berres

Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions

In this paper, two existing one-dimensional mathematical models, one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions, are combined into a model of continuous separation of polydisperse mixtures. This model can be written as a first-order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable. This application motivates the extension of the Kurganov-Tadmor central difference scheme to systems with discontinuous flux. The new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws. These additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step, or solved exactly, that is, the discontinuity parameters are kept constant (with respect to time). Numerical examples and an L1 error study show that the Kurganov-Tadmor scheme with first-order in time discretization produces spurious oscillations, whereas its semi-discrete version, discretized by a second-order Runge-Kutta scheme, produces good results. The scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved. Numerical examples illustrate the application to separation of polydisperse suspensions.

On gravity and centrifugal settling of polydisperse suspensions forming compressible sediments

Abstract A mathematical model describing both the hindered settling and the consolidation of suspensions with particles of different sizes and densities forming compressible sediments is presented. The specific new element is a centrifugal configuration, which gives rise to a non-constant body force. Within a range of angular velocities, the model can be reduced to one (radial) space dimension. The result is a system of second-order strongly degenerate parabolic–hyperbolic convection–diffusion equations. For the special subcase of suspensions of rigid spheres, which do not form compressible sediments and for which the effective solid stress can be assumed to vanish, these equations form a first-order system of conservation laws. A type analysis shows that these equations include hyperbolic, hyperbolic–parabolic, and hyperbolic–elliptic systems, depending on the sizes and densities of the solid particles. A numerical high-resolution central difference scheme for the hyperbolic and hyperbolic–parabolic models is applied to solve the model numerically, and thereby to simulate centrifugation of two polydisperse suspensions.

An adaptive finite-volume method for a model of two-phase pedestrian flow

A flow composed of two populations of pedestrians moving in different directions is modeled by a two-dimensional system of convection-diffusion equations. An efficient simulation of the two-dimensional model is obtained by a finite-volume scheme combined with a fully adaptive multiresolution strategy. Numerical tests show the flow behavior in various settings of initial and boundary conditions, where different species move in countercurrent or perpendicular directions. The equations are characterized as hyperbolic-elliptic degenerate, with an elliptic region in the phase space, which in one space dimension is known to produce oscillation waves. When the initial data are chosen inside the elliptic region, a spatial segregation of the populations leads to pattern formation. The entries of the diffusion-matrix determine the stability of the model and the shape of the patterns.

Centrifugal Settling of Flocculated Suspensions: A Sensitivity Analysis of Parametric Model Functions

The centrifugal settling of a flocculated suspension in a rotating tube can be modeled by a strongly degenerate parabolic partial differential equation whose coefficients depend on two material-specific model functions, namely, the hindered settling function and the effective solid stress function. These model functions are usually given by certain nonlinear algebraic expressions that involve a small number of parameters. The present work is related to the problem of determining these parameters for a given material. This problem of parameter identification consists in minimizing the distance between observed and simulated concentration profiles by successively varying the parameters employed for the simulation, starting from an initial guess. The feasibility and robustness of this procedure, which does not necessarily lead to a unique solution, decisively depends on the sensitivity of the solution of the direct problem to the different scalar parameters. These sensitivities are evaluated by a series of numerical experiments. It turns out that the model is extremely sensitive to the choice of the so-called critical concentration marking the transition between hindered settling and compression. Moreover, the robustness of the parameter identification method depends significantly on whether intermediate (i.e., transient) or stationary concentration profiles are used for identification.

Neumann Problems for Quasi-linear Parabolic Systems Modeling Polydisperse Suspensions

We discuss the well-posedness of a class of Neumann problems for n x n quasi-linear parabolic systems arising from models of sedimentation of polydisperse suspensions in engineering applications. This class of initial-boundary value problems includes the standard (zero-flux) Neumann condition in the limit as a positive perturbation parameter theta goes to 0. We call, in general, the problem associated with theta \ge 0 the theta-flux Neumann problem. The Neumann boundary conditions, although natural and usually convenient for integration by parts, are nonlinear and couple the different components of the system. An important aspect of our analysis is a time stepping procedure that considers linear boundary conditions for each time step in order to circumvent the difficulties arising from the nonlinear coupling in the original boundary conditions. We prove the well-posedness of the theta-flux Neumann problems for theta > 0 and obtain a solution of the standard (zero-flux) Neumann problem as the limit for the...

Mathematical Models for the Sedimentation of Suspensions

Mathematical models for sedimentation processes are needed in numerous industrial applications for the description, simulation, design and control of solid-liquid separation processes of suspensions. The first simple but complete model describing the settling of a monodisperse suspension of small rigid spheres is the kinematic sedimentation model due to Kynch [93], which leads to a scalar nonlinear conservation law. The extension of this model to flocculated suspensions, pressure filters, polydisperse suspensions and continuously operated clarifier-thickener units give rise to a variety of time-dependent partial differential equations with intriguing non-standard properties. These properties include strongly degenerate parabolic equations, free boundary problems, strongly coupled systems of conservation laws which may fail to be hyperbolic, and conservation laws with a discontinuous flux. This contribution gives an overview of the authors’ research that has been devoted to the mathematical modeling of solid-liquid separation, the existence and uniqueness analysis of these equations, the design and convergence analysis of numerical schemes, and the application to engineering problems. Extensions to other applications and general contributions to mathematical analysis are also addressed.

Simulation of an epidemic model with nonlinear cross-diffusion

A spatially two-dimensional epidemic model is formulated by a reaction-diffusion system. The spatial pattern formation is driven by a cross-diffusion corresponding to a non-diagonal, uppertriangular diffusion matrix. Whereas the reaction terms describe the local dynamics of susceptible and infected species, the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation. have been proposed to study pattern formation induced by cross-diffusion (Ni 2004, Bendahmane et al. 2009b, Tian et al. 2010). In addition to a fundamental existence proof for general reactiondiffusion systems (Crandall et al. 1987), there are several approaches to analyze reaction-diffusion equations with one single ”cross-diffusion” that lead to a system with upper triangular diffusion matrix (Badraoui 2006, Daddiouaissa 2008). The structure of an upper triangular diffusion matrix has also been utilized in the existence analysis for systems of convection-diffusion equations with both Dirich-let and Neumann boundary conditions (see e.g. Frid & Shelukhin 2004, Frid & Shelukhin 2005, Berres et al. 2006). Besides numerous contributions to the development of numerical methods to solve reaction-diffusion equations in related contexts (Wong 2008, Phongthanapanich & Dechaumphai 2009), convergence proofs of associated finite volume schemes (Bendahmane & Sepúlveda 2009, Andreianov et al. 2011) and finite element formulations (Galiano et al. 2003, Barrett & Blowey 2004) have been provided. This contribution is a condensed version of Berres & Ruiz-Baier 2011. The goal is, on the one hand, to generate pattern formation in an epidemic model by a cross-diffusion term, and, on the other hand, to prevent blow-up by a nonlinear limitation of the cross-diffusion. These assumptions are designed to qualitatively reflect psychological behavior. The cross-diffusion term has the interpretation that the susceptible population moves away from increasing gradients of the

A macroscopic model for crossing pedestrian streams

We model a macroscopic multi-pedestrian flow by a convection-diffusion equation. The convection corresponds to a movement towards a strategic direction whereas the diffusion corresponds to a tactical movement that avoids jams. Different populations moving in different directions are represented by different phases.

A simulation model for two-phase pedestrian flow

We introduce a two‐dimensional macroscopic model for the simulation of pedestrian flow based on a multiphase approach, which is well suited for the case of intersecting streams. A flow composed of two populations of pedestrians moving in different directions is modeled by a two‐dimensional system of convection‐diffusion equations. The model is intended to be the basic step of a more complex simulation model.

Mixed-type systems of convection-diffusion equations modeling polydisperse sedimentation

Models for the sedimentation of polydisperse suspensions of particles differing in size or density include strictly hyperbolic or mixed hyperbolic-elliptic systems of first-order conservation laws and strongly degenerate parabolic-hyperbolic systems of second-order convection-diffusion equations. The type depends on the properties of the solid particles. We present a summary of recent analyses of such systems and a numerical simulation of the settling of a bidisperse suspension.