Mathematical Aspects of Numerical Solution of Hyperbolic Systems: 1st Edition (Hardback) book cover

Mathematical Aspects of Numerical Solution of Hyperbolic Systems

1st Edition

By A.G. Kulikovskii, N.V. Pogorelov, A. Yu. Semenov

Chapman and Hall/CRC

560 pages

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Hardback: 9780849306082
pub: 2000-12-21
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Description

This important new book sets forth a comprehensive description of various mathematical aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important mechanical applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics (MHD), shallow water, and solid dynamics equations. This treatment provides-for the first time in book form-a collection of recipes for applying higher-order non-oscillatory shock-capturing schemes to MHD modelling of physical phenomena.

The authors also address a number of original "nonclassical" problems, such as shock wave propagation in rods and composite materials, ionization fronts in plasma, and electromagnetic shock waves in magnets. They show that if a small-scale, higher-order mathematical model results in oscillations of the discontinuity structure, the variety of admissible discontinuities can exhibit disperse behavior, including some with additional boundary conditions that do not follow from the hyperbolic conservation laws. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. The authors formulate several selection rules, which in some cases easily allow a correct, physically realizable choice.

This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes Mathematical Aspects of Numerical Solution of Hyperbolic Systems particularly valuable not only to those interested the development of numerical methods, but to physicists and engineers who strive to solve increasingly complicated nonlinear equations.

Reviews

"The book is a substantial addition to the existing literature… It will be of interest to students and researchers in fluid dynamics and continuum mechanics in various field of physics."

-European Mathematical Society Newsletter, No. 41 (September 2001)

" …this book…is as a sort of encyclopedia on numerical techniques applied to hyperbolic systems. Being free of, although important, mathematical and physical details, it allows the authors to focus the reader's attention on the core of numerics. The book is worthy of being in the library of everyone interested not only in numerical methods, but also in applied mathematics, mechanics, physics, and engineering, since the hyperbolic conservation laws are the basis of these areas of research."

-Applied Mathematics Review, Vol. 55, no. 3, May 2002

Table of Contents

HYPERBOLIC SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

Quasi-Linear Systems

Hyperbolic Systems

Mechanical Examples

Properties of Solutions

Disintegration of a Small Arbitrary Discontinuity

NUMERICAL SOLUTION OF QUASILINEAR HYPERBOLIC SYSTEMS

Introduction

Methods Based on the Exact Solution of the Riemann Problem

Methods Based on Approximate Riemann Problem Solvers

Generalized Riemann Problem

The Godunov Method of the Second Order

Multidimensional Schemes and their Stability Conditions

Reconstruction Procedures and Slope Limiters

Boundary Conditions for Hyperbolic Systems

Shock-Fitting Methods

Entropy Correction Procedures

Final Remarks

GAS DYNAMIC EQUATIONS

Systems of Governing Equations

The Godunov Method for Gas Dynamic Equations

Exact Solution of the Riemann Problem

Approximate Riemann Problem Solvers

Shock-Fitting Methods

Stationary Gas Dynamics

Solar Wind-Interstellar Medium Interaction

SHALLOW WATER EQUATIONS

System of Governing Equations

The Godunov Method for Shallow Water Equations

Exact Solution of the Riemann Problem

Results of Numerical Analysis

Approximate Riemann Problem Solvers

Stationary Shallow Water Equations

MAGNETOHYDRODYNAMIC EQUATIONS

MHD System in the Conservation-Law Form

Classification of MHD Discontinuities

Evolutionary MHD Shocks

High-Resolution Numerical Schemes for MHD Equations

Shock-Capturing Approach and Nonevolutionary Solutions in MHD

Strong background Magnetic Fields

Elimination of Numerical Magnetic Charge

Solar Wind Interaction with the Magnetized Interstellar Medium

SOLID DYNAMICS EQUATIONS

System of Governing Equations

CIR Method for the Calculation of Solid Dynamics Problems

CIR Method for Studying the Dynamics of Thin Shells

NONCLASSICAL DISCONTINUITIES AND SOLUTIONS OF HYPERBOLIC SYSTEMS

Evolutionary Conditions in Nonclassical Cases

Structure of Fronts. Additional Boundary Conditions on the Fronts

Behavior of the Hugoniot Curve in the Vicinity of Jouget Points and Nonuniqueness of Solutions of Self-Similar Problems

Nonlinear Small-Amplitude Waves in Anisotropic Elastic Media

Electromagnetic Shock Waves in Ferromagnets

Shock Waves in Composite Materials

Longitudinal Nonlinear Waves in Elastic Rods

Ionization Fronts in a Magnetic Field

Discussion

BIBLIOGRAPHY

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT021000
MATHEMATICS / Number Systems