Mathematical Aspects of Numerical Solution of Hyperbolic Systems
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Summary
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.
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
Editorial 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
