Thesis Summary:

In this dissertation, numerical schemes for the approximate solution of problems of multi-species kinematic flow models with (possibly) strongly degenerate diffusion terms are introduced and an analysis of a new diffusively corrected multi-species traffic flow model is presented. In particular, new numerical schemes for multi-species traffic flow models and polydisperse sedimentation problems are proposed. The thesis has the following aims. The first aim of this thesis is to demonstrate that an adaptive mesh refinement algorithm saves computational resources in simulations of polydisperse sedimentation model. The implementation of this adaptive technique is applied to two state-of-the-art high resolution shock capturing techniques. The second goal of this thesis is to demonstrate that implicit-explicit Runge-Kutta schemes efficiently generate a numerical solution of multi-species kinematic flow models with strongly degenerate diffusive term. Theses schemes consist in combining an explicit Runge-Kutta scheme for the time integration of the convective part with an implicit one for the diffusive part. To solve the highly nonlinear and non-smooth system that arises in the implicit discretization, it is proposed to regularize the diffusion coefficients and to apply the Newton-Raphson method with suitable globalization techniques. The CFL condition for the numerical scheme obtained is less severe than for an explicit treatment of the diffusive term. The third goal of this thesis is to propose a diffusively corrected multi-class Lighthill-WhithamRichards traffic model with anticipation lengths and reaction times. We analyse the stability of this diffusively corrected model under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature. Finally, we propose to introduce a new class of two-step numerical schemes for the multi-class Lighthill-Whitham-Richards traffic model. The new class of schemes combines in the first step the solutions of equations in Lagrangian coordinates and in a second step, a transport equation is solved to remap the solution to the original coordinates. The new schemes are referred to as “Lagrangian-remap” (LR) schemes. In the second step, two different strategies are considered. One strategy for LR schemes incorporates recent anti-diffusive techniques for transport equations. The corresponding subclass of LR schemes are named “Lagrangian-anti-diffusive-remap” (L-AR) schemes. The second strategy consists in handling the remap step by Glimm-like random sampling, which gives rise to a statistically conservative “Lagrangian-random sampling” (L-RS) scheme. The LR schemes for the MCLWR model are supported by a partial analysis of the L-AR schemes for N = 1, which are total variation diminishing (TVD) under a suitable CFL condition and therefore produce numerical solutions that converge to a weak solution, and by numerical examples for both L-AR and L-RS subclasses of schemes.

ISI Publications from the Thesis

Raimund BüRGER, Christophe CHALONS, Luis M. VILLADA: Anti-diffusive and random-sampling Lagrangian-remap schemes for the multi-class Lighthill-Whitham-Richards traffic model. SIAM Journal on Scientific Computing, vol. 35, 6, pp. 1341-1368, (2013)

Raimund BüRGER, Pep MULET, Luis M. VILLADA: A diffusively corrected multiclass Lighthill-Whitham-Richards traffic model with anticipation lengths and reaction times. Advances in Applied Mathematics and Mechanics, vol. 5, 5, pp. 728-758, (2013) .

Raimund BüRGER, Pep MULET, Luis M. VILLADA: Regularized nonlinear solvers for IMEX methods applied to diffusively corrected multi-species kinematic flow models. SIAM Journal on Scientific Computing, 35 (3), pp. B751-B777, (2013).

Raimund BüRGER, Pep MULET, Luis M. VILLADA: Spectral WENO schemes with adaptive mesh refinement for models of polydisperse sedimentation. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), vol. 93, 6-7, pp. 373-386, (2013).