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Conservation laws are the mathematical expression of the principles of conservation and provide effective and accurate predictive models of our physical world. Although intense research activity during the last decades has led to substantial advances in the development of powerful computational methods for conservation laws, their solution remains a challenge and many questions are left open; thus it is an active and fruitful area of research. Numerical Methods for Conservation Laws: From Analysis to Algorithms : offers the first comprehensive introduction to modern computational methods and their analysis for hyperbolic conservation laws, building on intense research activities for more than four decades of development ; discusses classic results on monotone and finite difference/finite volume schemes, but emphasizes the successful development of high-order accurate methods for hyperbolic conservation laws ; addresses modern concepts of TVD and entropy stability, strongly stable Runge-Kutta schemes, and limiter-based methods before discussing essentially nonoscillatory schemes, discontinuous Galerkin methods, and spectral methods ; explores algorithmic aspects of these methods, emphasizing one- and two-dimensional problems and the development and analysis of an extensive range of methods ; includes MATLAB software with which all main methods and computational results in the book can be reproduced ; and demonstrates the performance of many methods on a set of benchmark problems to allow direct comparisons. Code and other supplemental material will be available online at publication. This book is intended for graduate students in computational mathematics and researchers seeking a comprehensive introduction to modern methods for solving conservation laws. Students and researchers in applied sciences and engineering will benefit from the book's emphasis on algorithmic aspects of complex algorithms. The text also includes extensive references which allows researchers to pursue advanced research and results.

Conservation laws (Physics) --- Lois de conservation (physique) --- MATLAB (logiciel)

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Shock waves --- Conservation laws (Physics) --- Differential equations, Hyperbolic --- Ondes de choc --- Lois de conservation (Physique) --- Numerical solutions

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Au cours des trente dernières années, les progrès de la physique théorique ont été étroitement corrélés à ceux des mathématiques, en particulier dans le domaine de la théorie des cordes. Ce livre porte à la fois sur les développements de la physique et des mathématiques contemporaines. La conception de l'ouvrage est extrêmement originale : l'auteur ne se contente pas d'exposer sa vision des théories scientifiques, mais il appuie son argumentation en lui donnant un aspect ludique grâce à des énigmes intrigantes de mathématiques et de physique qui défient parfois l'intuition. Par exemple, si l'on rallonge d'un mètre une ceinture entourant la Terre à l'Equateur, quelle est la taille de l'objet qu'on peut glisser sous cette ceinture ? Après une revue des progrès récents dans notre compréhension de l'Univers (symétries et modèle standard des particules, trous noirs et cosmologie, etc.), l'auteur développe dans l'avant dernier chapitre le lien intime entre physique et mathématiques, en introduisant en particulier les notions de dualité et de symétrie miroir en théorie des cordes. Cet ouvrage s'adresse aux étudiants de physique et de mathématiques et à toute personne curieuse des développements récents de la science. Edward Witten, lauréat de la médaille Fields de mathématiques, écrit à propos de cet ouvrage : "Ce livre est un voyage fascinant et original dans les idées les plus avancées en physique et en mathématiques. Il est illustré par des énigmes élémentaires et amusantes. Le lecteur y trouvera de quoi apprendre et de quoi s'amuser"

Physics --- Mathematics --- Symmetry (Physics) --- Conservation laws (Physics) --- Conservation laws (Mathematics) --- Univers --- Mathématiques --- Physique --- Symétrie (physique) --- Lois de conservation (physique) --- Lois de conservation (mathématiques) --- Universe

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Symmetry (Physics) --- Conservation laws (Physics) --- Differential invariants --- Physics --- Mathematics --- Symétrie (Physique) --- Lois de conservation (Physique) --- Invariants différentiels --- Physique --- Mathématiques --- History --- Histoire --- Noether, Emmy, --- Influence. --- Noether's theorem. --- Mathématiques --- Symétrie (physique) --- Calcul des variations. --- History. --- Histoire. --- Noether, Emmy --- Symétrie (physique)

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The seeds of Continuum Physics were planted with the works of the natural philosophers of the eighteenth century, most notably Euler; by the mid-nineteenth century, the trees were fully grown and ready to yield fruit. It was in this envi­ ronment that the study of gas dynamics gave birth to the theory of quasilinear hyperbolic systems in divergence form, commonly called "hyperbolic conserva­ tion laws"; and these two subjects have been traveling hand-in-hand over the past one hundred and fifty years. This book aims at presenting the theory of hyper­ bolic conservation laws from the standpoint of its genetic relation to Continuum Physics. Even though research is still marching at a brisk pace, both fields have attained by now the degree of maturity that would warrant the writing of such an exposition. In the realm of Continuum Physics, material bodies are realized as continuous media, and so-called "extensive quantities", such as mass, momentum and energy, are monitored through the fields of their densities, which are related by balance laws and constitutive equations. A self-contained, though skeletal, introduction to this branch of classical physics is presented in Chapter II. The reader may flesh it out with the help of a specialized text on the subject.

Champs [Théorie des ] (Physique) --- Conservatiewetten (Fysica) --- Conservation [Lois de ] --- Conservation laws (Physics) --- Differentiaalvergelijkingen [Hyperbolische ] --- Differential equations [Hyperbolic] --- Equations différentielles hyperboliques --- Field theory (Physics) --- Lois de conservation (Physique) --- Veldtheorie (Fysica) --- Differential equations, Hyperbolic. --- Conservation laws (Physics). --- Field theory (Physics). --- Partial differential equations. --- Thermodynamics. --- Mechanics. --- Partial Differential Equations. --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Chemistry, Physical and theoretical --- Mechanics --- Heat --- Heat-engines --- Partial differential equations