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Approximation theory --- Differential equations, Hyperbolic --- Fourier analysis --- Théorie de l'approximation --- Analyse de Fourier --- Numerical solutions --- Fourier Analysis --- -Approximation theory --- Analysis, Fourier --- Mathematical analysis --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Hyperbolic differential equations --- Differential equations, Partial --- Approximation theory. --- Fourier analysis. --- Numerical solutions. --- Théorie de l'approximation --- Numerical analysis --- Differential equations, Hyperbolic - Numerical solutions

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Systems governed by non-linear differential equations are of fundamental importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics for slowly moving monopoles. This picture clarifies the important notion of solitons, which has attracted much attention in recent years. The soliton idea bridges the gap between the concepts of "fields" and "particles," and is here explored in a fully three-dimensional context. While the background and motivation for the work comes from physics, the presentation is mathematical.This book is interdisciplinary and addresses concerns of theoretical physicists interested in elementary particles or general relativity and mathematicians working in analysis or geometry. The interaction between geometry and physics through non-linear partial differential equations is now at a very exciting stage, and the book is a contribution to this activity.Originally published in 1988.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Magnetic monopoles --- Solitons. --- Geometry. --- Differential equations, Hyperbolic --- Solitons --- Géométrie --- Mathematics. --- Numerical solutions. --- -Geometry --- -Solitons --- Numerical solutions --- Mathematics --- 530.1 --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- Monopoles, Magnetic --- Electromagnetism --- Magnetic pole --- Euclid's Elements --- Hyperbolic differential equations --- Differential equations, Partial --- Basic principles of physics --- 530.1 Basic principles of physics --- Géométrie --- Geometry --- Numerical analysis --- Differential equations, Hyperbolic - - Numerical solutions --- Magnetic monopoles - - Mathematics

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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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Finite volume method. --- Conservation laws (Mathematics). --- -Finite volume method --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Finite volume method --- Volumes finis, Méthodes de --- Conservation laws (Mathematics) --- Differential equations, Hyperbolic --- 514.7 --- 519.6 --- 681.3 *G18 --- Numerical analysis --- 514.7 Differential geometry. Algebraic and analytic methods in geometry --- Differential geometry. Algebraic and analytic methods in geometry --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Numerical solutions --- Numerical solutions. --- Lois de conservation (Mathématiques) --- Differential equations, Hyperbolic - Numerical solutions.