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This volume contains the texts of the four series of lectures presented by B.Cockburn, C.Johnson, C.W. Shu and E.Tadmor at a C.I.M.E. Summer School. It is aimed at providing a comprehensive and up-to-date presentation of numerical methods which are nowadays used to solve nonlinear partial differential equations of hyperbolic type, developing shock discontinuities. The most effective methodologies in the framework of finite elements, finite differences, finite volumes spectral methods and kinetic methods, are addressed, in particular high-order shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon a-posteriori error analysis.

Differential equations, Hyperbolic --- Differential equations, Nonlinear --- Numerical solutions --- Congresses --- Differential equations [Hyperbolic] --- Differential equations [Nonlinear ] --- Partial differential equations. --- Numerical analysis. --- Thermodynamics. --- Computational intelligence. --- Partial Differential Equations. --- Numerical Analysis. --- Computational Intelligence. --- Chemistry, Physical and theoretical --- Dynamics --- Mechanics --- Physics --- Heat --- Heat-engines --- Quantum theory --- Mathematical analysis --- Partial differential equations --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Differential equations, Hyperbolic - Numerical solutions - Congresses --- Differential equations, Nonlinear - Numerical solutions - Congresses

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Optimal Shape Design is concerned with the optimization of some performance criterion dependent (besides the constraints of the problem) on the "shape" of some region. The main topics covered are: the optimal design of a geometrical object, for instance a wing, moving in a fluid; the optimal shape of a region (a harbor), given suitable constraints on the size of the entrance to the harbor, subject to incoming waves; the optimal design of some electrical device subject to constraints on the performance. The aim is to show that Optimal Shape Design, besides its interesting industrial applications, possesses nontrivial mathematical aspects. The main theoretical tools developed here are the homogenization method and domain variations in PDE. The style is mathematically rigorous, but specifically oriented towards applications, and it is intended for both pure and applied mathematicians. The reader is required to know classical PDE theory and basic functional analysis.

Mathematical optimization. --- Structural optimization --- Mathematics. --- Mathematical optimization --- Optimalisation mathématique --- Wiskundige optimisatie --- Mathematics --- Structural optimization - Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Calculus of variations. --- Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- 517.1 Mathematical analysis --- Mathematical analysis