K. Hvistendahl Karlsen

Numerical methods for the simulation of the settling of flocculated suspensions

For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic‐parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e. conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.

Numerical simulation of the settling of polydisperse suspensions of spheres

The extension of Kynchs kinematical theory of ideal suspensions to polydisperse suspensions of spheres leads to a nonlinear system of conservation laws for the volumetric concentration of each species. In this work, we consider particle species different in sizes and densities, including the buoyant case. We show that modern shock-capturing numerical schemes for the solution of systems of conservation laws can be employed as an efficient tool for the simulation of the settling and separation of polydisperse suspensions. This is demonstrated by comparison with published experimental and theoretical results and by simulating some hypothetical configurations. Particular attention is focused on the emergence of rarefaction waves.

On some upwind difference schemes for the phenomenological sedimentation-consolidation model

In one space dimension, the phenomenological sedimentation-consolidation model reduces to an initial-boundary value problem (IBVP) for a nonlinear strongly degenerate convection-diffusion equation with a non-convex, time-dependent flux function. The frequent assumption that the effective stress of the sediment layer is a function of the local solids concentration only which vanishes below a critical concentration value causes the model to be of mixed hyperbolic-parabolic nature. Consequently, its solutions are discontinuous and entropy solutions must be sought. In this paper, first a (short) guided visit to the mathematical (entropy solution) framework in which the well-posedness of this and a related IBVP can be established is given. This also includes a short discussion of recent existence and uniqueness results for entropy solutions of IBVPs. The entropy solution framework constitutes the point of departure from which numerical methods can be designed and analysed. The main purpose of this paper is to present and demonstrate several finite-difference schemes which can be used to correctly simulate the sedimentation-consolidation model in civil and chemical engineering and in mineral processing applications, i.e., conservative schemes satisfying a discrete entropy principle. Here, finite-difference schemes of upwind type are considered. To some extent, also stability and convergence properties of the proposed schemes are discussed. Performance of the proposed schemes is demonstrated by simulation of two cases of batch settling and one of continuous thickening of flocculated suspensions. The numerical examples focus on a detailed error study, an illustration of the effect of varying the initial datum, and on simulation of practically important thickener operations, respectively.

Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations

The extension of Kynchs kinematic theory of sedimentation of monodisperse suspensions to polydisperse mixtures leads to a nonlinear system of conservation laws for the volume fractions of each species. In this paper, we show that a second-order central (Riemann-solver-free) scheme for the solution of systems of conservation laws can be employed as an efficient tool for the simulation of the settling and the separation of polydisperse suspensions. This is demonstrated by comparison with a published experimental study of the settling of a bidisperse suspension. In addition, we compare the prediction of the one-dimensional kinematic sedimentation model with a three-dimensional particle-scale simulation.

A fast marching method for reservoir simulation

We present a fast marching level set method for reservoir simulation based on a fractional flow formulation of two-phase, incompressible, immiscible flow in two or three space dimensions. The method uses a fast marching approach and is therefore considerably faster than conventional finite difference methods. The fast marching approach compares favorably with a front tracking method as regards both efficiency and accuracy. In addition, it maintains the advantage of being able to handle changing topologies of the front structure.

The corrected operator splitting approach applied to a nonlinear advection-diffusion problem

So-called corrected operator splitting methods are applied to a 1-D scalar advection-diffusion equation of Buckley-Leverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piece wise linear finite elements at each new time level. To obtain correct structure of shock fronts independently of the size of the time step, a dynamically defined residual flux term is grouped with diffusion. Different test problems are considered, and the methods are compared with respect to accuracy and runtime. Finally, we extend the corrected operator splitting to 2-D equations by means of dimensional splitting, and we apply it to a Buckley-Leverett type problem including gravitational effects.

Dimensional Splitting with Front Tracking and Adaptive Grid Refinement

Front tracking in combination with dimensional splitting is analyzed as a numerical method for scalar conservation laws in two space dimensions. An analytic error bound is derived, and convergence rates based on numerical experiments are presented. Numerical experiments indicate that large CFL numbers can be usedwithoutlossofaccuracyforawiderangeofproblems. Anewmethodforgridrefinementisintroduced. The method easily allows for dynamical changes in the grid, using, for instance, the total variation in each gridcellasacriterionforintroducingneworremovingexistingrefinements. Severalnumericalexamplesare included, highlighting the features of the numerical method. A comparison with a high-resolution method confirms that dimensional splitting with front tracking is a highly viable numerical method for practical computations. c 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 627{648, 1998

Front Tracking and Operator Splitting for Nonlinear Degenerate Convection-Diffusion Equations

We describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convection—diffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The conservation law is solved by a front tracking method, while the diffusion equation is here solved by a finite difference scheme. The numerical methods are unconditionally stable in the sense that the (splitting) time step is not restricted by the spatial discretization parameter. The strategy is designed to handle all combinations of convection and diffusion (including the purely hyperbolic case). Two numerical examples are presented to highlight the features of the methods, and the potential for parallel implementation is discussed.

A Front Tracking Method for Conservation Laws with Boundary Conditions

A front tracking method is used to construct weak solutions to scalar conservation laws with two kinds of boundary conditions — Dirichlet conditions and a novel zero flux (or no-flow) condition. The construction leads to an efficient numerical method. The main feature of the scheme is that there is no stability condition correlating the spatial and temporal discretization parameters. The analysis uses the traditional method of proving compactness via Helly’s theorem as well as the more modern concept of measure valued solutions. Three numerical examples are presented.

A study of the modelling error in two operator splitting algorithms for porous media flow

Operator splitting methods are often used to solve convection–diffusion problems of convection dominated nature. However, it is well known that such methods can produce significant (splitting) errors in regions containing self sharpening fronts. To amend this shortcoming, corrected operator splitting methods have been developed. These approaches use the wave structure from the convection step to identify the splitting error. This error is then compensated for in the diffusion step. The main purpose of the present work is to illustrate the importance of the correction step in the context of an inverse problem. The inverse problem will consist of estimating the fractional flow function in a one‐dimensional saturation equation.