This thesis studies the behavior of the Eulerian scheme, with "Wave Confinement" (WC), when propagating periodic waves. WC is a recently developed method that was derived from the scheme "vorticity confinement" used in fluid mechanics, and it efficiently solves the linear wave equation. This new method is applicable for numerous simulations such as radio wave propagation, target detection, cell phone and satellite communications. The WC scheme adds a nonlinear term to the discrete wave equation that adds stability with negative and positive diffusion, conserves integral quantities such as total amplitude and wave speed, and it allows wave propagation over long distances with minimal numerical diffusion, which contrasts to other numerical methods where wave propagation is affected by numerical dissipation. Previous studies have shown that WC propagates short pulses/surfaces as thin nonlinear solitary waves. In this thesis, a one-dimensional (1D) periodic wave is propagated by WC using the advection and wave equations. For the advection equation, the parameters and the initial condition (IC) used in WC are analyzed to establish for which conditions the method can be implemented. When the IC is a positive periodic wave, the converged solution consists of a series of hyperbolic secants where the number of cycles of the IC represents the number of hyperbolic secants. Waves with varying signs are analyzed by changing the wave confinement term. For this case, the converged solution is a series of positive and negative hyperbolic secants where each hyperbolic secant is represented by half cycle of the IC. For the wave equation, parameters and different IC's are studied to determine when WC is feasible. For positive periodic waves, the converged solution retains its sinusoidal shape and does not converge to a series of hyperbolic secants. The waves with varying signs, however, converge to a series of hyperbolic secants as seen for the advection equation. WC is stable for various periodic waves for both advection and wave equations, which shows WC is useful for numerically propagating periodic waveforms. Convergence depends on the wave number of the IC and on the parameters (convection speed, positive diffusion, negative diffusion) used in WC.

Cette thèse est articulée autour de deux facettes de l’étude des équations auxdérivées partielles. Dans une première partie, on étudie la stabilité des solutionspériodiques pour des lois de conservation. On démontre d’abord la stabilité asymptotiquedans L1 des solutions périodiques de lois de conservation scalaires et inhomogènes.On montre ensuite un résultat de stabilité structurelle des roll-waves. Plusprécisément, on montre que les solutions périodiques d’un système hyperbolique sansviscosité sont limites des solutions du problème avec viscosité, quand le terme deviscosité tend vers 0. Dans une deuxième partie, on s’intéresse à un système d’équationsaux dérivées partielles issu de la biologie : le modèle de Patlak-Keller-Segelen dimension 2 ; il décrit les phénomènes de chimiotactisme. Pour ce modèle, onconstruit un schéma de type volume fini, ce qui permet d’approcher la solution touten gardant certaines propriétés du système : positivité, conservation de la masse,estimation d’énergie. / This thesis is organized around two aspects of the study of partial differentialequations. In a first part, we study the stability of periodic solutions for conservationlaws. First, we prove asymptotic L1-stability of periodic solutions of scalarinhomogeneous conservation laws. Then, we show a result on structural stability ofroll-waves. More precisely, we prove that periodic solutions of a hyperbolic systemwithout viscosity are the limits of the solutions of the problem with viscosity, as theviscous term tends to 0. In a second part, we study a system of partial differentialequations derived from biology: the model of Patlak-Keller-Segel in dimension 2, describingthe phenomena of chemotaxis. For this model, we construct a finite-volumescheme, which approaches the solution while keeping some properties of the system:positivity, conservation of mass, energy estimate.

In this dissertation, we are interested in the dynamics of spatially distributed populations. In particular, we focus on persistence conditions and minimal traveling periodic wave speeds for stage-structured populations in heterogeneous landscapes. The model includes structured populations of two age groups, juveniles and adults, in patchy landscapes. First, we present a stage-structured population model, where we divide the population into pre-reproductive and reproductive stages. We assume that all parameters of the two age groups are piecewise constant functions in space. We derive explicit formulas for population persistence in a single-patch landscape and in heterogeneous habitats. We find the critical size of a single patch surrounded by a non-lethal matrix habitat. We derive the dispersion relation for the juveniles-adults model in homogeneous and heterogeneous landscapes. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest. We apply sensitivity and elasticity formulas to the critical size of a single-patch landscape and to the minimal traveling wave speed in a homogeneous landscape. Secondly, we use asymptotic techniques to find an explicit formula for the traveling periodic wave speed and to calculate the spread rates for structured populations in heterogeneous landscapes. We illustrate the power of the homogenization method by comparing the dispersion relation and the resulting minimal wave speeds for the approximation and the exact expression. We find an excellent agreement between the fully heterogeneous speed and the homogenized speed, even though the landscape period is on the same order as the diffusion coefficients and not as small as the formal derivation requires. We also generalize this work to the case of structured populations of n age groups. Lastly, we use a finite difference method to explore the numerical solutions for the juveniles-adults model. We compare numerical solutions to analytic solutions and explore population dynamics in non-linear models, where the numerical solution for the time-dependent problem converges to a steady state. We apply our theory to study various aspects of marine protected areas (MPAs). We develop a model of two age groups, juveniles and adults, in which only adults can be harvested and only outside MPAs, and recruitment is density dependent and local inside MPAs and fishing grounds. We include diffusion coefficients in density matching conditions at interfaces between MPAs and fishing grounds, and examine the effect of fish mobility and bias movement on yield and fish abundance. We find that when the bias towards MPAs is strong or the difference in diffusion coefficients is large enough, the relative density of adults inside versus outside MPAs increases with adult mobility. This observation agrees with findings from empirical studies.