Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes

The generic structure of solutions of initial value problems of hyperbolic- elliptic systems, also called mixed systems, of conservation laws is not yet fully understood. One reason for the absence of a core well-posedness theory for these equations is the sensitivity of their solutions to the structure of a parabolic regularization when attempting to single out an admissible solution by the vanishing viscosity approach. There is, however, theoretical and numerical evidence for the appearance of solutions that exhibit persistent oscillations, so-called oscillatory waves, which are (in general, measure-valued) solutions that emerge from Riemann data or slightly perturbed constant data chosen from the interior of the elliptic region. To capture these solutions, usually a fine computational grid is required. In this work, a version of the multiresolution method applied to a WENO scheme for systems of conservation laws is proposed as a simulation tool for the efficient computation of solutions of oscillatory wave type. The hyperbolic-elliptic 2×2 systems of conservation laws considered are a prototype system for three-phase flow in porous media and a system modeling the separation of a heavy-buoyant bidisperse suspension. In the latter case, varying one scalar parameter produces elliptic regions of different shapes and numbers of points of tangency with the borders of the phase space, giving rise to different kinds of oscillation waves

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

On diffusively corrected multispecies kinematic flow models

This presentation provides a survey of some recent results related to efficient numerical methods for the numerical solution of a class of convection-diffusion systems that arise as one-dimensional models of the flow of one (disperse) substance through a continuous fluid. Applications include the settling of polydisperse suspensions of solid particles in a viscous fluid, multiclass vehicular traffic under the effect of anticipation distances and reaction times, the settling of dipersions and emulsions, and chromatography. In many of these applications the system becomes strongly degenerate. For the numerical solution, this fact poses a number of difficulties whose partial solution will be addressed. For instance, it is well known that implicit-explicit (IMEX) numerical scheme that are based on discretizing the convective and diffusive parts are a potentially suitable tool to avoid the severe time step limitation associated with fully explicit discretization. However, their implementation relies on the efficient numerical solution of the nonlinear systems of algebraic equations arising from the discretization which can not be achieved by standard Newton-Raphson techniques when the diffusion coefficients are discontinuous. A combined smoothing and line search technique solves the problem of solving the corresponding nonlinearly implicit equations. Alternatively, this problem can be avoided by the construction of so-called linearly implicit methods that are slightly less accurate, but noticeably more efficient than their nonlinearly implicit counterparts. The main collaborators in this research are Pep Mulet (Universitat de Valencia, Spain) and Luis Miguel Villada (Universidad del Bío-Bío, Concepción, Chile).Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Density dependent diffusion models for the interaction of particle ensembles with boundaries

The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing i) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ii) the interaction of flocks (of fish or birds) with boundaries and iii) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.Comment: 25 pages, 9 figure

On boundary conditions for multidimensional sedimentation-consolidation processes in closed vessels

The two-phase flow of a flocculated suspension in a closed settling vessel with inclined walls is investigated within the phenomenological theory of sedimentation-consolidation processes. We formulate possible wall boundary conditions and use these conditions to derive spatially one-dimensional field equations for planar flows and flows which are symmetric with respect to the vertical axis. For both kinds of flows we assume a general geometry of the sedimentation vessel and include the study of a compressible sediment layer. We analyze the special cases of a conical vessel, a roof-shaped vessel and a vessel with parallel inclined walls. The case of a small initial time and a large time for the final consolidation state leads to explicit expressions for the flow fields. From a mathematical point of view, the resulting initial-boundary value problems are well posed and can be solved numerically by a simple adaptation of one of the newly developed numerical schemes for strongly degenerate convection-diffusion problems. However, from a physical point of view, both the analytical and numerical results rise doubts concerning the validity of the general field equations. In particular, the strongly reduced form of the linear momentum balance seems to be an oversimplification. Included in our discussion as a special case are the Kynch theory and well-known analyses of sedimentation in vessels with inclined walls within the framework of kinematic waves, which exhibit similar shortcomings

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

Hyperbolic Conservation Laws

[no abstract available

Stability and asymptotic analysis of a fluid-particle interaction model

We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models