Paulo Amorim

On the Numerical Integration of Scalar Nonlocal Conservation Laws

We study a rather general class of 1D nonlocal conservation laws from a numerical point of view. First, following [F. Betancourt, R. Burger, K.H. Karlsen and E.M. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885], we define an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are led to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.

Modeling ant foraging: A chemotaxis approach with pheromones and trail formation

We consider a continuous mathematical description of a population of ants and simulate numerically their foraging behavior using a system of partial differential equations of chemotaxis type. We show that this system accurately reproduces observed foraging behavior, especially spontaneous trail formation and efficient removal of food sources. We show through numerical experiments that trail formation is correlated with efficient food removal. Our results illustrate the emergence of trail formation from simple modeling principles.

CONVERGENCE OF NUMERICAL SCHEMES FOR SHORT WAVE LONG WAVE INTERACTION EQUATIONS

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schrodinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.

Analysis of a chemotaxis system modeling ant foraging

In this paper we analyze a system of PDEs recently introduced in [P. Amorim, {\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail formation}], in order to describe the dynamics of ant foraging. The system is made of convection-diffusion-reaction equations, and the coupling is driven by chemotaxis mechanisms. We establish the well-posedness for the model, and investigate the regularity issue for a large class of integrable data. Our main focus is on the (physically relevant) two-dimensional case with boundary conditions, where we prove that the solutions remain bounded for all times. The proof involves a series of fine \emph{a priori} estimates in Lebesgue spaces.

Convergence of a finite difference method for the KdV and modified KdV equations with

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in

L^2

L^2

data

), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.

A nonlinear model describing a short wave–long wave interaction in a viscoelastic medium

In this paper we introduce a system coupling a nonlinear Schrodinger equation with a system of viscoelasticity, modeling the interaction between short and long waves, acting for instance on media such as plasmas or polymers. We prove the existence and uniqueness of local (in time) strong solutions and the existence of global weak solutions for the corresponding Cauchy problem. In particular we extend previous results in [Nohel et. al., Commun. Part. Diff. Eq., 13 (1988)] for the quasilinear system of viscoelasticity. We finish with some numerical computations to illustrate our results.

Sharp estimates for periodic solutions to the Euler–Poisson–Darboux equation

We establish sharp estimates for distributional solutions to the Euler-Poisson- Darboux equation posed in a periodic domain. These equations are highly singular, and setting the Cauchy problem requires a precise understanding of the nature of the sin- gularities that may arise in weak solutions. We consider initial data in a space of func- tions with fractional derivatives such that weak solutions are solely integrable, and we derive sharp continuous dependence estimates for solutions to the initial-value problem. Our results strongly depend on a key parameter arising in the Euler-Poisson-Darboux equation.

An ant navigation model based on Weber’s law

We analyze an ant navigation model based on Weber’s law, where the ants move across a pheromone landscape sensing the area using two antennae. The key parameter of the model is the angle