Thesis Summary:

This thesis deals with the mathematical and numerical analysis of two models that describe the behavior of multiple species from partial differential equations. In particular, a system of conservation laws with a discontinuous flow function and a reaction-diffusion system coupled with elliptic equations are considered, modeling traffic flow problems that distinguish between free-congested ow and the dynamics of populations that interact with chemotaxis. The main contents of this thesis is structured as follows: In Chapter 1, we construct a numerical scheme that is similar to the one proposed by [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722], by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of N conservation laws for N driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented. In Chapter 2, we formulate a reaction-dffusion system to describe three interacting species within the Hastings-Powell (HP) food chain structure with chemotaxis produced by three chemicals. We construct a finite volume (FV) scheme for this system, and in combination with the non-negativity and a priori estimates for the discrete solution, the existence of a discrete solution of the FV scheme is proved. It is shown that the scheme converges to the corresponding weak solution of the model. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L1 compactness argument. Finally, numerical tests illustrate the model and the behavior of the FV scheme. In Chapter 3, we consider a mathematical model for the spatio-temporal evolution of three biological species in a food chain model consisting of two competitive preys and one predator with intra-specific competition. The species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting reaction-dffusion system consists of three parabolic equations along with three elliptic equations describing the behavior of the chemical substance. First, the local existence of nonnegative solutions is proved, then we provide uniform estimates in Lebesgue spaces which lead to boundedness and the global well-posedness for the system. Finally we report and discuss some numerical simulation.

ISI Publications from the Thesis

Raimund BüRGER, Christophe CHALONS, Rafael ORDOñEZ, Luis M. VILLADA: A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks and Heterogeneous Media, vol. 16, 2, pp. 187-219, (2021).

Raimund BüRGER, Rafael ORDOñEZ, Mauricio SEPúLVEDA, Luis M. VILLADA: Numerical analysis of a three-species chemotaxis model. Computers & Mathematics with Applications, vol. 80, pp. 183-203, (2020).