Magali Ribot

A fluid dynamics model of the growth of phototrophic biofilms

A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing.

A hyperbolic model of chemotaxis on a network: a numerical study

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

Asymptotic High Order Mass-Preserving Schemes for a Hyperbolic Model of Chemotaxis

We introduce a new class of finite difference schemes for approximating the solutions to an initial-boundary value problem on a bounded interval for a one-dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to nonconstant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. Special care is needed to deal with boundary conditions to avoid harmful loss of mass. Convergence results are proved for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.

Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis

We consider a model of chemotaxis with finite speed of propagation based on Cattaneos law in two space dimensions. For this system we present a global existence theorem for the smooth solutions to the Cauchy problem. This result is obtained using estimates of the Green functions of the linearized operators. We illustrate the behavior of solutions by some numerical simulations.

A Numerical Comparison Between Degenerate Parabolic and Quasilinear Hyperbolic Models of Cell Movements Under Chemotaxis

We consider two models which were both designed to describe the movement of eukaryotic cells responding to chemical signals. Besides a common standard parabolic equation for the diffusion of a chemoattractant, like chemokines or growth factors, the two models differ for the equations describing the movement of cells. The first model is based on a quasilinear hyperbolic system with damping, the other one on a degenerate parabolic equation. The two models have the same stationary solutions, which may contain some regions with vacuum. We first explain in details how to discretize the quasilinear hyperbolic system through an upwinding technique, which uses an adapted reconstruction, which is able to deal with the transitions to vacuum. Then we concentrate on the analysis of asymptotic preserving properties of the scheme towards a discretization of the parabolic equation, obtained in the large time and large damping limit, in order to present a numerical comparison between the asymptotic behavior of these two models. Finally we perform an accurate numerical comparison of the two models in the time asymptotic regime, which shows that the respective solutions have a quite different behavior for large times.

A Numerical Scheme for a Hyperbolic Relaxation Model on Networks

In this article, we consider a simple hyperbolic relaxation system on networks which models the movement of fibroblasts on an artificial scaffold. After proving the uniqueness of stationary solutions with a given total mass, we present an adapted numerical scheme which takes care of boundary conditions and display some numerical tests.

STABILITY, CONVERGENCE AND ORDER OF THE EXTRAPOLATIONS OF THE RESIDUAL SMOOTHING SCHEME IN ENERGY NORM

The Residual Smoothing Scheme is a numerical method which consists in preconditioning at each time step the method of lines. In this paper, RSS is defined and analyzed in an abstract linear parabolic case, i.e. for an abstract ordinary differential equation of the form with A a self-adjoint non negative operator, and it can be written where B is a preconditioner of A. We show that RSS is stable, convergent and of order one in energy norm. We also prove that its kth Richardsons extrapolation is stable and of order k.