Thesis Summary:

High order numerical methods are developed in this work to approximate the solution of multi-species kinematic flow models and convection-diffusion systems. Specifically, numerical schemes are proposed for vehicular traffic models, polydisperse sedimentation and liquid chromatography. The following are the objectives that are pursued in this thesis. The figures with the exception of Figure 5.1 are own preparation. The first objective of this thesis is to show the advantages in terms of efficiency that the PVM (Polynomial Viscosity Matrix) method presents when simulating multispecies kinematic flow models under a suitable selection of the approximate viscosity of Roe’s numerical flux by using different types of Gaussian integration. The second objective of this thesis is to propose a new PVM method which does not present oscillations when the viscosity matrix is approximated by using fourth degree polynomials, as for example in the Masliyah-Lockett-Bassoon (MLB) model when taking a large number of species and the second eigenvalue is very close to zero with respect to the largest eigenvalue. This model arises from the continuity and linear momentum balance equations for the solid species and the fluid, for equal-density particles the velocities of this model are given by (3.5). The third objective of this work is to use the techniques mentioned above to approximate the convective term of convection-diffusion systems in the simulation of sedimentation of droplets of different diameters dispersed in a viscous fluid. It is proposed to use Runge-Kutta Implicit-Explicit schemes to obtain an efficient solution of these convection-diffusion systems. Finally we use Runge-Kutta Implicit-Explicit schemes with the method of lines approach to efficiently obtain approximate solutions of a liquid chromatography model, which is a powerful tool for separation of complex mixtures. Keywords. PVM (Polynomial viscosity matrix), Roe matrix, Conservation law systems, Multi-species kinematic flow models, Polydispersed sedimentation, Vehicle traffic models.

ISI Publications from the Thesis

Raimund BüRGER, Pep MULET, Lihki RUBIO, Mauricio SEPúLVEDA: Linearly implicit-explicit schemes for the equilibrium dispersive model of chromatography. Applied Mathematics and Computation, vol. 317, pp. 172-186, (2018).

Raimund BüRGER, Pep MULET, Lihki RUBIO: Polynomial viscosity methods for multispecies kinematic flow models. Numerical Methods for Partial Differential Equations, vol. 32, 4, pp. 1265-1288, (2016).