1st Edition

The Two-Dimensional Riemann Problem in Gas Dynamics

By Jiequan Li, Tong. Zhang, Shuli Yang Copyright 1998
    310 Pages
    by Chapman & Hall

    The Riemann problem is the most fundamental problem in the entire field of non-linear hyperbolic conservation laws. Since first posed and solved in 1860, great progress has been achieved in the one-dimensional case. However, the two-dimensional case is substantially different. Although research interest in it has lasted more than a century, it has yielded almost no analytical demonstration. It remains a great challenge for mathematicians.
    This volume presents work on the two-dimensional Riemann problem carried out over the last 20 years by a Chinese group. The authors explore four models: scalar conservation laws, compressible Euler equations, zero-pressure gas dynamics, and pressure-gradient equations. They use the method of generalized characteristic analysis plus numerical experiments to demonstrate the elementary field interaction patterns of shocks, rarefaction waves, and slip lines. They also discover a most interesting feature for zero-pressure gas dynamics: a new kind of elementary wave appearing in the interaction of slip lines-a weighted Dirac delta shock of the density function.
    The Two-Dimensional Riemann Problem in Gas Dynamics establishes the rigorous mathematical theory of delta-shocks and Mach reflection-like patterns for zero-pressure gas dynamics, clarifies the boundaries of interaction of elementary waves, demonstrates the interesting spatial interaction of slip lines, and proposes a series of open problems. With applications ranging from engineering to astrophysics, and as the first book to examine the two-dimensional Riemann problem, this volume will prove fascinating to mathematicians and hold great interest for physicists and engineers.

    Preface
    Preliminaries
    Geometry of Characteristics and Discontinuities
    Riemann Solution Geometry of Conservation Laws
    Scalar Conservation Laws
    One-Dimensional Scalar Conservation Laws
    The Generalized Characteristic Analysis Method
    The Four-Wave Riemann Problem
    Mach-Reflection-Like Configuration of Solutions
    Zero-Pressure Gas Dynamics
    Characteristics and Bounded Discontinuities
    Simultaneous Occurrence of Two Blowup Mechanisms
    Delta-Shocks, Generalized Rankine-Hugoniot Relations and Entropy Conditions
    The One-Dimensional Riemann Problem
    The Two-Dimensional Riemann Problem
    Riemann Solutions as the Limits of Solutions to Self-Similar Viscous Systems
    Pressure-Gradient Equations of the Euler System
    The Pme-Dimensional Riemann Problem
    Characteristics, Discontinuities, Elementary Waves, and Classifications
    The Existence of Solutions to a Transonic Pressure-Gradient Equation in an Elliptic Region with Degenerate Datum
    The Two-Dimensional Riemann Problem and Numerical Solutions
    The Compressible Euler Equations
    The Concepts of Characteristics and Discontinuities
    Planar Elementary Waves and Classification
    PSI Approach to Irrotational Isentropic Flow
    Analysis of Riemann Solutions and Numerical Results
    Two-Dimensional Riemann Solutions with Axisymmetry
    References
    Author Index

    Biography

    Li, Jiequan; Zhang, Tong.; Yang, Shuli

    "A complete and rigorous study…"
    -Mathematical Reviews
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