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Numerical simulation of compressible, inviscid time-dependent flow is a major branch of computational fluid dynamics. Its primary goal is to obtain accurate representation of the time evolution of complex flow patterns, involving interactions of shocks, interfaces, and rarefaction waves. The Generalized Riemann Problem (GRP) algorithm, developed by the authors for this purpose, provides a unifying 'shell' which comprises some of the most commonly used numerical schemes of this process. This 2003 monograph gives a systematic presentation of the GRP methodology, starting from the underlying mathematical principles, through basic scheme analysis and scheme extensions (such as reacting flow or two-dimensional flows involving moving or stationary boundaries). An array of instructive examples illustrates the range of applications, extending from (simple) scalar equations to computational fluid dynamics. Background material from mathematical analysis and fluid dynamics is provided, making the book accessible to both researchers and graduate students of applied mathematics, science and engineering.

Fluid dynamics. --- Riemann-Hilbert problems. --- Hilbert-Riemann problems --- Riemann problems --- Boundary value problems --- Dynamics --- Fluid mechanics

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In singularity theory and algebraic geometry the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the theory of systems of linear differential equations one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. In the theory of homomorphic foliations there appear the Ecalle-Voronin-Martinet-Ramis moduli. On the other hand, there is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. All this is presented in this book, underlining the unifying role of the monodromy group. The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. The book contains a lot of results which are usually spread in many sources. Readers can quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature.

Monodromy groups. --- Riemann-Hilbert problems. --- Hilbert-Riemann problems --- Riemann problems --- Boundary value problems --- Group theory --- Algebra. --- Algebraic topology. --- Functions of complex variables. --- Differential Equations. --- Functions, special. --- Algebraic Topology. --- Functions of a Complex Variable. --- Ordinary Differential Equations. --- Special Functions. --- Special functions --- Mathematical analysis --- 517.91 Differential equations --- Differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Topology --- Mathematics --- Differential equations. --- Special functions.

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Painleve equations --- Riemann-Hilbert problems --- Differential equations, Nonlinear --- Asymptotic theory --- Painlevé equations. --- Riemann-Hilbert problems. --- 517.9 --- Asymptotic theory in nonlinear differential equations --- Asymptotic expansions --- Hilbert-Riemann problems --- Riemann problems --- Boundary value problems --- Equations, Painlevé --- Functions, Painlevé --- Painlevé functions --- Painlevé transcendents --- Transcendents, Painlevé --- Asymptotic theory. --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Painlevé equations --- Differential equations, Nonlinear - Asymptotic theory