Um estudo de métodos Lagrangiano-Euleriano para problemas hiperbólicos e leis de balanço

Orientador: Eduardo Cardoso de AbreuDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O foco deste trabalho consiste em estudar e aplicar o esquema Lagrangiano-Euleriano para leis de conservação hiperbólicas escalares e leis de balanço escalares. Portanto, apresentamos ao longo do estudo algumas defnições básicas e exemplos de conceitos fundamentais relacionados. O esquema Lagrangiano-Euleriano tem como objetivo não ser dependente de uma estrutura particular do termo de fonte. Um conjunto de experimentos numéricos - escalar e sistemas - para leis de conservação hiperbólica e leis de balanço são apresentados para ilustrar o desempenho do método, como a equação de Burgers, a equação de Buckley-Leverett, Equações de águas rasas e o ?uxo trifásico não-miscivel. Enfm, aplicamos a abordagem do esquema Lagrangiano-Euleriano para lei de conservação hiperbólica com ?uxo não local e condições iniciais de medida como a Gaussiana e condição inicial oscilatória. Sempre que for possível, os resultados numéricos são comparados com soluções aproximadas ou soluções exatasAbstract: The focus of this work consists on the study and on the application of the Lagrangian-Eulerian scheme for scalar hyperbolic conservation laws and scalar hyperbolic balance laws. For this purpose, we present some basic defnitions and examples of fundamental concepts related throughout the study. The Lagrangian-Eulerian scheme is aimed to be not dependent on a particular structure of the source term. Furthermore, a set of representative numerical experiments - scalar and system - of hyperbolic conservation laws and balance laws are presented to illustrate the performance of the method such as the Burgers¿ equation, the Buckley-Leverett¿s equation, Shallow water equations and the Immiscible three-phase ?ow. For completeness, we apply the Lagrangian-Eulerian framework to hyperbolic conservation laws with nonlocal ?ux, with measure initial datas such as Guassian initial data and Oscillatory initial data. Whenever possible, we make a comparison between the numerical results and accurate approximate solutions or exact solutionsMestradoMatematica AplicadaMestre em Matemática Aplicada132128/2016-0CNP

A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data

International audienceThis article is dedicated to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions, or more generally nonconstant monotonic bounded functions, as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multitype version of the sticky particle dynamics and obtain existence of global weak solutions by compactness. We then derive a $L^p$ stability estimate on the particle system uniform in the number of particles. This allows to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all orders, which encompasses the classical $L^1$ estimate and generalises to diagonal systems the results by Bolley, Brenier and Loeper [J. Hyperbolic Differ. Equ., 2005] in the scalar case. Our results are obtained without any smallness assumption on the variation of the data, and only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic