Made available in DSpace on 2016-06-02T20:27:38Z (GMT). No. of bitstreams: 1 3160.pdf: 1167165 bytes, checksum: d968da16f55a4ed2bbefe667a52d3d80 (MD5) Previous issue date: 2010-08-17 / Financiadora de Estudos e Projetos / In this work, we studied the construction of weak solutions for some systems conservation laws and for a scalar conservation laws For the p-systems relativistic we use the Wave-front tracking based on Bressan, when the total variation of the initial data is locally bounded. We construct is somewhat more simpler than in methods studied by other authors. For the equation of Aw-Rascle we apply the method of Glimm and the method Wave-front tracking, we consider the appearence of the vacuum, when the total variation of the initial data is locally bounded. For the scalar conservation law we study the construction of admissible weak solutions using the Glimm method and the poligonal approximations. / Neste trabalho, estudamos a construção de soluções fracas para alguns sistemas de leis de conservação e para uma lei de conservação escalar. Para o p-sistema relativístico aplicamos o método Wave-front tracking baseado em Bressan, quando a variação total do dado inicial é localmente limitada. O nosso método de construção é um pouco mais simples que os métodos estudados por outros autores. Para as equações de Aw-Rascle aplicamos o método de Glimm e o método Wave-front tracking permitindo ou não o aparecimento de vácuo nas soluções aproximadas, quando a variação total do dado inicial é localmente limitada. Para a lei de conservação escalar estudamos a construção de soluções fracas admissíveis utilizando o método de Glimm e a aproximação poligonal.

<p>U doktorskoj disertaciji posmatrani su Rimanovi problemi kod strogo i slabo hiperboličnih&nbsp;nelinearnih sistema PDJ. U uvodu je predstavljena jednačina zakona održanja u jednoj prostornoj&nbsp;dimenziji i definisani su Ko&scaron;ijevi i Rimanovi problemi. U drugoj glavi, date su osnovne osobine&nbsp;nelinearnih hiperboličnih zakona održanja, uvedeni supojmovi stroge hiperboličnosti i slabog re&scaron;enja&nbsp;zakona održanja. Definisani su Rankin-Igono i entropijski uslovi kao i op&scaron;te re&scaron;enje Rimanovog problema&nbsp;(za dovoljno male početne uslove). U trećoj glavi detaljno je obja&scaron;njena Glimova diferencna &nbsp;&scaron;ema. Metod&nbsp;praćenja talasa predstavljen je u četvrtoj glavi. Pokazano je da se ovom metodom, za dovoljno male&nbsp;početne uslove, dobija stabilno i jedinstveno re&scaron;enje koje u svakom vremenu ima ograničenu totalnu&nbsp;varijaciju. U petoj glavi, posmatrana je jednačina protoka izentropnog gasa u Lagranžovim koordinatama.&nbsp;Uz pretpostavku da je početni uslov ograničen i da ima ograničenu totalnu varijaciju, pokazano je da&nbsp;Ko&scaron;ijev problem ima jedinstveno slabo re&scaron;enje ako je totalna varijacija početnog uslova pomnožena sa &nbsp;0&lt;&epsilon;&lt;&lt; 1 dovoljno mala. Slabo re&scaron;enjedobijeno je metodom praćenja talasa. U glavi &scaron;est ispitana je&nbsp;interakcija dva delta talasa koji su posmatrani kao specijalna vrsta shadowtalasa. U glavi sedam,&nbsp;pokazano je da za proizvoljno velike početne uslove, re&scaron;enje Rimanovog problema jednodimenzionalnog&nbsp;Ojlerovog zakona održanja gasne dinamikepostoji, daje jedinstveno i entropijski dopustivo, uz drugačiju<br />ocenu snaga elementarnih talasa. Data je numerička verifikacija interakcije dva delta talasa kori&scaron;ćenjem&nbsp;metode praćenja talasa.</p> / <p>In this doctoral thesis, Riemann problems for strictly and weakly nonlinear hyperbolic PDE&nbsp;systems were observed. In the introduction, conservation laws in one spatial dimension were presented&nbsp;and the Cauchy and Riemann problems were defined. In the second chapter, the basic properties of&nbsp;nonlinear hyperbolic conservation laws were intorduced, as well as the terms such as strictly hyperbolic&nbsp;system and weak solution of conservation law. Also, Rankine -Hugoniot and entropy conditions were<br />introduced and the general solution to the Riemann problem (for sufficiently small initial conditions) were&nbsp;defined. Glimm&rsquo;s difference scheme was explained in the third chapter. The wave front tracking method&nbsp;was introduced in the fourth chapter. It was shown that, using this method, for sufficiently small initial&nbsp;conditions, it could be obtained a unique solution with bounded total variation for t &ge;0. In the fifth&nbsp;chapter, the Euler equations for isentropic fluid inLagrangian coordinates were observed. Under the&nbsp;assumption that the initial condition was bounded and had bounded total variation, it was shown that the&nbsp;Cauchy problem had a weak unique solution, provided that the total variation of initial condition&nbsp;multiplied by 0&lt;&epsilon;&lt;&lt;1 was sufficiently &nbsp;small. Weak solution was obtained by applying the wave front&nbsp;tracking method. In the sixth chapter, the interaction of two delta shock waves were examined. Delta&nbsp;shock waves were regarded as special kind of shadowwaves. In the chapter seven, it was shown that for&nbsp;arbitrarily large initial conditions, solution to the Riemann problem of one-dimensional Euler&nbsp;conservation laws of gas dynamics existed, it was unique and admissible. New bounds on the strength of&nbsp;elementary waves in the wave front tracking algorithm were given. The numerical verification of two&nbsp;delta shock waves interaction via wave front tracking method was given at the end of the thesis.</p>

Dans cette thèse, on considère plusieurs problèmes issus de la modélisation macroscopique des mouvements de foule. Le premier modèle consiste en une loi de conservation avec un flux discontinu, le second est un système mixte hyperbolique-elliptique et le dernier est une équation non-locale. D'abord, on utilise le modèle de Hughes une dimension pour décrire l'évacuation d'un couloir avec deux sorties. Ce modèle couple une loi de conservation avec un flux discontinu à une équation eikonale. On implémente la méthode de suivi de fronts, qui traite explicitement le comportement de la solution non-classique au point de rebroussement, afin d'obtenir des solutions de référence. Elles serviront à tester numériquement la convergence de schémas aux volumes finis classiques. Ensuite, on modélise le croisement de deux groupes marchant dans des directions opposées avec un système de lois de conservation mixte hyperbolique-elliptique dont le flux dépend des deux densités. Le système perd son hyperbolicité pour certainement valeurs de densité. On assiste à l'apparition d'oscillations persistantes mais bornées, ce qui conduit à la reformulation du problème associé dans le cadre des mesures de probabilités. Finalement, on étudie un modèle non-local de trafic piétonnier en deux dimensions. Le modèle consiste en une loi de conservation dont le flux dépend d'une convolution de la densité. Avec ce modèle, on résout un problème d'optimisation pour une évacuation d'une salle avec une méthode de descente, évaluant l'impact du calcul explicite du gradient de la fonction coût avec la méthode de l'état adjoint plutôt que son approximation par différences finies. / In this thesis, we consider nonclassical problems brought out by the macroscopic modeling of pedestrian flow. The first model consists of a conservation law with a discontinuous flux, the second is a mixed hyperbolic-elliptic system of conservation laws and the last one is a nonlocal equation. In the first chapter, we use the Hughes model in one space-dimension to represent the evacuation of a corridor with two exits. The model couples a conservation law with discontinuous flux to an eikonal equation. We implement the wave front tracking scheme, treating explicitly the solution nonclassical behavior at the turning point, to provide a reference solution, which is used to numerically test the convergence of classical finite volume schemes. In the second chapter, we model the crossing of two groups of pedestrians walking in opposite directions with a system of conservation laws whose flux depends on the two densities. This system loses its hyperbolicity for certain density values. We assist to the rising of persistent but bounded oscillations, that lead us to the recast of the problem in the framework of measure-valued solutions. Finally we study a nonlocal model of pedestrian flow in two space-dimensions. The model consists of a conservation law whose flux depends on a convolution of the density. With this model, we solve an optimization problem for a room evacuation with a descent method, evaluating the impact of the explicit computation of the cost function gradient with the adjoint state method rather than approximating it with finite differences.