On diffusively corrected multispecies kinematic flow models

Abstract

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).